Download Lectures on Quantum Gravity and Black Holes

Lectures on Quantum Gravity and Black Holes
Thomas Hartman
Cornell University
Please email corrections and suggestions to: [email protected]
Abstract These are the lecture notes for a one-semester graduate course on black
holes and quantum gravity. We start with black hole thermodynamics, Rindler space,
Hawking radiation, Euclidean path integrals, and conserved quantities in General Relativity. Next, we rediscover the AdS/CFT correspondence by scattering fields off nearextremal black holes. The final third of the course is on AdS/CFT, including correlation
functions, black hole thermodynamics, and entanglement entropy. The emphasis is on
semiclassical gravity, so topics like string theory, D-branes, and super-Yang Mills are
discussed only very briefly.
Course Cornell Physics 7661, Spring 2015
Prereqs This course is aimed at graduate students who have taken 1-2 semesters of
general relativity (including: classical black holes, Penrose diagrams, and the Einstein
action) and 1-2 semesters of quantum field theory (including: Feynman diagrams, path
integrals, and gauge symmetry.) No previous knowledge of quantum gravity or string
theory is necessary.
1
Contents
1 The problem of quantum gravity
8
1.1
Gravity as an effective field theory
. . . . . . . . . . . . . . . . . . . .
8
1.2
Quantum gravity in the Ultraviolet . . . . . . . . . . . . . . . . . . . .
15
1.3
Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2 The Laws of Black Hole Thermodynamics
19
2.1
Quick review of the ordinary laws of thermodynamics . . . . . . . . . .
19
2.2
The Reissner-Nordstrom Black Hole . . . . . . . . . . . . . . . . . . . .
20
2.3
The 1st law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4
The 2nd law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.5
Higher curvature corrections . . . . . . . . . . . . . . . . . . . . . . . .
27
2.6
A look ahead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3 Rindler Space and Hawking Radiation
30
3.1
Rindler space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.2
Near the black hole horizon . . . . . . . . . . . . . . . . . . . . . . . .
31
3.3
Periodicity trick for Hawking Temperature . . . . . . . . . . . . . . . .
32
3.4
Unruh radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.5
Hawking radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4 Path integrals, states, and operators in QFT
41
4.1
Transition amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.2
Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.3
Cutting the path integral . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.4
Euclidean vs. Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.5
The ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2
4.6
Vacuum correlation functions . . . . . . . . . . . . . . . . . . . . . . .
46
4.7
Density matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.8
Thermal partition function . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.9
Thermal correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5 Path integral approach to Hawking radiation
54
5.1
Rindler Space and Reduced Density Matrices . . . . . . . . . . . . . . .
54
5.2
Example: Free fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.3
Importance of entanglement . . . . . . . . . . . . . . . . . . . . . . . .
59
5.4
Hartle-Hawking state
63
. . . . . . . . . . . . . . . . . . . . . . . . . . .
6 The Gravitational Path Integral
68
6.1
Interpretation of the classical action . . . . . . . . . . . . . . . . . . . .
68
6.2
Evaluating the Euclidean action . . . . . . . . . . . . . . . . . . . . . .
69
7 Thermodynamics of de Sitter space
75
7.1
Vacuum correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
7.2
The Static Patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
7.3
Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
8 Symmetries and the Hamiltonian
83
8.1
Parameterized Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
8.2
The ADM Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
8.3
Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
8.4
Other conserved charges . . . . . . . . . . . . . . . . . . . . . . . . . .
89
8.5
Asymptotic Symmetry Group . . . . . . . . . . . . . . . . . . . . . . .
90
8.6
Example: conserved charges of a rotating body . . . . . . . . . . . . . .
93
9 Symmetries of AdS3
97
3
9.1
Exercise: Metric of AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . .
97
9.2
Exercise: Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
9.3
Exercise: Conserved charges . . . . . . . . . . . . . . . . . . . . . . . . 100
10 Interlude: Preview of the AdS/CFT correspondence
102
10.1 AdS geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10.2 Conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.3 Statement of the AdS/CFT correspondence . . . . . . . . . . . . . . . 104
11 AdS from Near Horizon Limits
107
11.1 Near horizon limit of Reissner-Nordstrom . . . . . . . . . . . . . . . . . 107
11.2 6d black string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
12 Absorption Cross Sections of the D1-D5-P
114
12.1 Gravity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
13 Absorption cross section from the dual CFT
121
13.1 Brief Introduction to 2d CFT . . . . . . . . . . . . . . . . . . . . . . . 121
13.2 2d CFT at finite temperature . . . . . . . . . . . . . . . . . . . . . . . 125
13.3 Derivation of the absorption cross section . . . . . . . . . . . . . . . . . 127
13.4 Decoupling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
14 The Statement of AdS/CFT
131
14.1 The Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
14.2 Example: IIB Strings and N = 4 Super-Yang-Mills . . . . . . . . . . . 133
14.3 General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
14.4 The Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . 136
15 Correlation Functions in AdS/CFT
4
138
15.1 Vacuum correlation functions in CFT . . . . . . . . . . . . . . . . . . . 138
15.2 CFT Correlators from AdS Field Theory . . . . . . . . . . . . . . . . . 140
15.3 Quantum corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
16 Black hole thermodynamics in AdS5
143
16.1 Gravitational Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . 144
16.1.1 Schwarzschild-AdS . . . . . . . . . . . . . . . . . . . . . . . . . 144
16.1.2 Thermal AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
16.1.3 Hawking-Page phase transition . . . . . . . . . . . . . . . . . . 149
16.1.4 Large volume limit . . . . . . . . . . . . . . . . . . . . . . . . . 151
16.2 Confinement in CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
16.3 Free energy at weak and strong coupling . . . . . . . . . . . . . . . . . 153
17 Eternal Black Holes and Entanglement
157
17.1 Thermofield double formalism . . . . . . . . . . . . . . . . . . . . . . . 157
17.2 Holographic dual of the eternal black hole . . . . . . . . . . . . . . . . 159
17.3 ER=EPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
17.4 Comments in information loss in AdS/CFT
. . . . . . . . . . . . . . . 163
17.5 Maldacena’s information paradox . . . . . . . . . . . . . . . . . . . . . 163
17.6 Entropy in the thermofield double . . . . . . . . . . . . . . . . . . . . . 165
18 Introduction to Entanglement Entropy
166
18.1 Definition and Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
18.2 Geometric entanglement entropy . . . . . . . . . . . . . . . . . . . . . . 169
18.3 Entropy Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
19 Entanglement Entropy in Quantum Field Theory
175
19.1 Structure of the Entanglement Entropy . . . . . . . . . . . . . . . . . . 176
5
19.2 Lorentz invariance
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
20 Entanglement Entropy and the Renormalization Group
180
20.1 The space of QFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
20.2 How to measure degrees of freedom . . . . . . . . . . . . . . . . . . . . 181
20.3 Entanglement proof of the c-theorem . . . . . . . . . . . . . . . . . . . 183
20.4 Entanglement proof of the F theorem . . . . . . . . . . . . . . . . . . . 185
21 Holographic Entanglement Entropy
187
21.1 The formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
21.2 Example: Vacuum state in 1+1d CFT . . . . . . . . . . . . . . . . . . 188
21.3 Holographic proof of strong subadditivity . . . . . . . . . . . . . . . . . 191
21.4 Some comments about HEE . . . . . . . . . . . . . . . . . . . . . . . . 192
22 Holographic entanglement at finite temperature
194
22.1 Planar limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
23 The Stress Tensor in 2d CFT
198
23.1 Infinitessimal coordinate changes . . . . . . . . . . . . . . . . . . . . . 198
23.2 The Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
23.3 Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
23.4 Operator product expansion . . . . . . . . . . . . . . . . . . . . . . . . 204
23.5 The Central Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
23.6 Casimir Energy on the Circle . . . . . . . . . . . . . . . . . . . . . . . 208
24 The stress tensor in 3d gravity
210
24.1 Brown-York tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
24.2 Conformal transformations and the Brown-Henneaux central charge . . 211
24.3 Casimir energy on the circle . . . . . . . . . . . . . . . . . . . . . . . . 212
6
25 Thermodynamics of 2d CFT
214
25.1 A first look at the S transformation . . . . . . . . . . . . . . . . . . . . 214
25.2 SL(2, Z) transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 217
25.3 Thermodynamics at high temperature
26 Black hole microstate counting
. . . . . . . . . . . . . . . . . . 219
221
26.1 From the Cardy formula . . . . . . . . . . . . . . . . . . . . . . . . . . 221
26.2 Strominger-Vafa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7
1
The problem of quantum gravity
These are lectures on quantum gravity. To start, we better understand clearly what
problem we are trying to solve when we say ‘quantum gravity.’ At low energies, the
classical action is
1
S=
16πGN
Z
√
−g (R − 2Λ + Lmatter ) .
(1.1)
Why not just quantize this action? The answer of course is that it is not renormalizable.
This does not mean it is useless to understand quantum gravity, it just means we have
to be careful about when it is reliable and when it isn’t. In this first lecture we will
consider gravity as a low-energy effective field theory,∗ see when it breaks down, and
make some general observations about what we should expect or not expect from the
UV completion.
1.1
Gravity as an effective field theory
The rules of effective field theory are:
1. Write down the most general possible action consistent with the symmetries;
2. Keep all terms up to some fixed order in derivatives;
3. Coefficients are fixed by dimensional analysis, up to unknown order 1 factors
(unless you have a good reason to think otherwise);
4. Do quantum field theory using this action, including loops;
5. Trust your answer only if the neglected terms in the derivative expansion are
much smaller than the terms you kept.
This works for renormalizable or non-renormalizable theories. Let’s follow the steps
for gravity. Our starting assumption is that nature has a graviton — a massless spin-2
field. This theory can be consistent only if it is diffeomorphism invariant.
∗
Donaghue gr-qc/9512024.
8
Counting metric degrees of freedom
This can be argued various ways∗ ; we’ll just count degrees of freedom. In 4D, a massless
particle has two degrees of freedom (2 helicities). Similarly, the metric gµν has†
10 components − 4 diffeos − 4 non-dynamical = 2 dof .
(1.2)
In D dimensions, we count dof of a massless particle by looking at how the particle
states transform under SO(D − 2), the group of rotations that preserve a null ray.‡ A
spin-2 particle transforms in the symmetric traceless tensor rep of SO(D − 2), which
has dimension 21 (D − 2)(D − 1) − 1 = 12 D(D − 3). Similarly, assuming diffeomorphism
invariance, the metric has
1
D(D
2
+ 1) − D − D = 21 D(D − 3)
(1.3)
degrees of freedom.
Note that in D = 3, the metric has no (local) dof. It turns out that it does have some
nonlocal dof; this will be useful later in the course.
Back to effective field theory: Steps 1 and 2, the derivative expansion
The only things that can appear in a diff-invariant Lagrangian for the metric are objects
built out of the Riemann tensor Rµνρσ and covariant derivatives ∇µ . Each Riemann
contains ∂∂g, so the derivative expansion is an expansion in the number of R’s and
∇’s. Up to 4th order in derivatives,
1
S=
16πGN
Z
√
−g −2Λ + R + c1 R2 + c2 Rµν Rµν + c3 Rµνρσ Rµνρσ + · · ·
(1.4)
So the general theory is the Einstein-Hilbert term plus higher curvature corrections.
We have ignored the matter terms Lmatter and matter-curvature couplings, like φR.
∗
See Weinberg QFT V1, section 5.9, and the discussion of the Weinberg-Witten theorem below,
and Weinberg Phys. Rev. 135, B1049 (1964).
†
In more detail: a 4x4 symmetric matrix has 10 independent components. In 4D we have 4
0
functions worth of diffeomorphisms, xµ → xµ (xµ ). And g̈0µ cannot appear in a 2nd order diff-invariant
equation of motion, so these components are non-dynamical. For more details, see the discussion of
gravitational waves in any introductory GR textbook, which should show that in transverse-traceless
gauge the linearized Einstein equation have two independent solutions (the ‘+’ and ‘×’ polarizations).
‡
See Weinberg QFT V1, section 2.5.
9
Step 3: Coefficients = scale of new physics
Coefficients should be fixed by dimensional analysis, up to O(1) factors. This doesn’t
work for the cosmological constant: experiment (ie the fact the universe is not Plancksized) indicates that Λ is unnaturally small. This is the cosmological constant problem.
We will just sweep this under the rug, take this fine-tuning as an experimental fact,
and proceed to higher order.
For these purposes let’s take the coordinates to have dimensions of length, so the metric
is dimensionless, and R has mass dimension 2. The action should be dimensionless
(since ~ = 1). Looking at the Einstein-Hilbert term, that means [GN ] = 2 − D, so in
terms of the Planck scale,
1
≡ (MP )D−2 .
GN
(1.5)
In D > 2, this term is not renormalizable. This means that the theory is strongly
coupled at the Planck scale. If we try to compute scattering amplitudes using Feynman
diagrams, we would find non-sensical, non-unitary answers for E & MP . The rules
of effective field theory tell us that we must include the R2 terms, with coefficients
c1,2,3 ∼ 1/MP2 . Higher curvature terms should also be included, suppressed by more
powers of MP . More generally, the rule is that these coefficients should be suppressed
by the scale of new physics, which we will call Ms . New physics must appear at or
below the Planck to save unitarity, so Ms < MP , but it’s possible that Ms MP . So
to allow for this possibility, we set
c1,2,3 ∼
1
.
Ms2
(1.6)
In string theory Ms would be the mass of excited string states:
Ms ∼
1
,
`s
c1,2,3 ∼ α0
(1.7)
where `s is the string length and α0 = `2s is the string tension. In this context the R2
and higher curvature terms in the action are called ‘stringy corrections.’
Steps 4 and 5: Do quantum field theory, but be careful what’s reliable
In this section to be concrete we will work in D = 4. To do perturbation theory (about
10
flat space) with the action (1.4), we set
gµν = ηµν +
1
hµν
MP
(1.8)
and expand in h, or equivalently in 1/MP . The factor of MP is inserted here so that
the quadratic action is canonically normalized; schematically, the perturbative action
looks like
Z
S∼
1
1
∂h∂h +
h∂h∂h + · · · + 2
MP
Ms
1
∂ h∂ h +
h∂ 2 h∂ 2 h + · · ·
MP
2
2
(1.9)
where the first terms come from expanding the Einstein action, and the other terms
come from the higher curvature corrections.∗ In curved space, the higher curvature
terms would also contribute to the terms like h∂h∂h since R ∼ const + ∂ 2 h + · · · .
Scattering and the strong coupling scale
As expected in a non-renormalizable theory, the perturbative expansion breaks down
at high energies. First consider the case where we set Ms = MP (or, we keep only the
Einstein term in the Lagrangian) and calculate the amplitude for graviton scattering
in perturbation theory:
√
= GN
√
GN + crosses +
+
√
(1.10)
+ ···
+
This is an expansion in the coupling constant
GN
GN = 1/MP ; but this is dimensionful, so
it must really be an expansion in E/MP . That is, each diagram contributes something
∗
√
The term written here comes from a term in the action
−g ∼ 1 + δg, R ∼ ∂ 2 δg, then rescale δg =
1
MP
h.
11
Ms
MP
2
R2 , where we pick off the terms
of order
E2
MP2
1+number of loops
.
(1.11)
So the strong coupling scale, where loop diagrams are the same size as tree diagrams,
is
Estrong ∼ MP .
(1.12)
Below the strong coupling scale, this is a perfectly good quantum theory. We can use
it to make reliable predictions about graviton-graviton scattering, including calculable
loop corrections.
If there are two different scales Ms and MP with Ms MP the situation is slightly
more complicated. The Einstein term is strongly coupled at the Planck scale, but
looking at (1.9), the higher curvature terms become strongly coupled at a lower scale
somewhere between Ms and MP ,
Estrong ∼ MPx Ms1−x
(1.13)
for some x ∈ (0, 1). (This is a known number that you can find by examining all the
diagrams.) It is important, however, that interactions in the higher curvature terms
still come with powers of 1/MP , so even in this case Estrong contains some factor of MP
(ie, x > 0): the theory is still weakly coupled at the scale of new physics, E ∼ Ms .
Classical corrections to the Newtonian potential and ghosts
Returning to the classical theory, consider for example the theory with just the first
term in (1.4), so c1 = (Ms )−2 and c2 = c3 = 0. The equations of motion derived from
this action are schematically
h +
1
Ms
2
h = 8πGN T .
(1.14)
Going to momentum space ∼ E 2 , so clearly the higher curvature term is negligible
at low energies E Ms . The propagator looks like
q2
1
1
1
= 2− 2
−2
4
+ Ms q
q
q + Ms2
12
(1.15)
The 2nd term looks like a massive field with mass Ms , but the wrong sign. It is a new,
non-unitary degree of freedom, or ‘ghost’, in the classical theory. Although the only
field is still the metric, it makes sense that we’ve added a degree of freedom because
we need more than 2 functions worth of initial data to solve the 4th order equation of
motion (1.14).
The ‘ghost’ should not bother us, because it appears at the scale Ms . This is the scale
of new physics where we should not trust our effective field theory anyway. And at
energies E Ms , the ghost has no effect on classical gravity. To see this, let’s compute
the classical potential between two massive objects. The first term in (1.15) gives the
Newtonian 1/r potential. The second term looks like a massive Yukawa force, so the
classical potential is
V (r) = −GN m1 m2
1 e−rMs
−
r
r
.
(1.16)
This tiny for distances r > 1/Ms .
Loop corrections to the Newtonian potential
The 2pt function has calculable, reliable quantum corrections:
+ ···
+
∼
1
1
1 a 4
q2 1
−
+
q
log
+ ···
q 2 Ms2 q 2 MP2
Λ2 q 2
(1.17)
The first two terms are the classical part,
q2
1
1
1
= 2 − 2 + ··· ,
−2
4
+ Ms q
q
Ms
(1.18)
and the log term is the loop diagram (including external legs!). a is an order 1 number
that can been calculated from this diagram, and we’ve dropped some terms to simplify
the discussion (see Donaghue for details). To calculate the attractive potential between
13
two stationary masses, we set the frequency to zero q 0 = 0 and go to position space,∗
Z
1
1
1
d ~q
+ 2 − 2 log ~q 2 + · · ·
2
~q
Ms
MP
3
ei~q·~x ∼
1
1
1
+ 2 δ(r) + 2 3
r Ms
MP r
(1.19)
This Fourier transform is just done by dimensional analysis (we’ve dropped numerical
coefficients). The first term is the classical Newtonian potential, the second term is
the classical higher-curvature correction, and the last term is a quantum correction.
The delta function does not matter at separated points; it is UV physics and does
not affect the potential. It came from the same physics as the Yukawa term e−rMs in
our discussion above— the difference is that the Yukawa term is the exact classical
contribution whereas the delta function comes from expanding out the propagator in
a derivative expansion.
The last term in (1.19) is a reliable prediction of quantum gravity, with small but
non-zero effects at low energies.
Does Ms = MP ?
So does Ms = MP , or is there a new scale Ms MP ? This is basically the question
of whether the new UV physics that fixes the problems of quantum gravity is weakly
coupled (Ms MP ) or strongly coupled (no new scale). Both options are possible,
and both are realized in different corners of string theory.
If we ask the analogous question about other effective field theories that exist in nature,
then sometimes the new physics is strongly coupled (for example, QCD as the UV
completion of the pion Lagrangian) and sometimes it’s weakly coupled (for example,
electroweak theory as the UV completion of Fermi’s theory of beta decay).
Breakdown
As argued above, the effective field theory breaks down at (or below) MP . It is conceivable that this is just a problem with perturbation theory, and that the theory makes
sense non-perturbatively, for example by doing the path integral on a computer. The
problem is that the theory is UV-divergent and must be renormalized; this means we
do not have the option of just plugging in a particular action, say just the Einstein term
∗
See for example Peskin and Schroeder section 4.7.
14
R√
−gR, and using this to define a quantum theory. We must include the full series of
higher curvature terms. Each comes with a coupling constant, so we have an infinite
number of tunable parameters and lose predictive power. The only way this theory
can make predictions is if these infinite number of running couplings flow under RG to
a UV fixed point with a finite number of parameters. This idea is called ‘asymptotic
safety,’ and although it’s a logical possibility there is little evidence for it. It is not
how other effective fields theories in nature (eg pions) have been UV completed.
1.2
Quantum gravity in the Ultraviolet
We have seen that at E MP , ordinary methods in quantum field theory can be
applied to gravity without any problems. So the real problem, of course, is how to
find a UV completion that reduces at low energy to the effective field theory we’ve just
described. This problem is unsolved, but there has been a lot of progress. The point of
this section is to make a few comments about what we should and should not expect
in a theory of quantum gravity, and to introduce the idea of emergent spacetime.
Quantum gravity has no local observables
Gauge symmetry is not a symmetry. It is a fake, a redundancy introduced by hand to
help us keep track of massless particles in quantum field theory. All physical predictions
must be gauge-independent.
In an ordinary quantum field theory without gravity, in flat spacetime, there two types
of physical observables that we most often talk about are correlation functions of
gauge-invariant operators hO1 (x1 ) · · · On (xn )i, and S-matrix elements. The correlators
are obviously gauge-independent. S-matrix elements are also physical, even though
electrons are not gauge invariant. The reason is that the states used to define the
S-matrix have particles at infinity, and gauge transformations acting at infinity are
true symmetries. They take one physical state to a different physical state — unlike
local gauge transformations, which map a physical state to a different description of
the same physical state.
In gravity, local diffeomorphisms are gauge symmetries. They are redundancies. This
means that local correlation functions like hO1 (x1 ) · · · On (xn )i are not gauge invariant,
15
and so they are not physical observables.∗ On the other hand, diffeomorphisms that
reach infinity (like, say, a global translation) are physical symmetries — taking states
in the Hilbert space to different states in the Hilbert space — so we get a physical
observable by taking the insertion points to infinity. This defines the S-matrix, so it is
sometimes said that ‘The S-matrix is the only observable in quantum gravity.’
This is not quite true, since there are also non-local physical observables. For example,
suppose we send in an observer from infinity, along a worldline xµ0 (τ ), with τ the
proper time along the path. Although the coordinate value xµ0 (τ ) depends on the
coordinate system, it unambiguously labels a physical point on the manifold; that is,
0
xµ0 (τ1 ) labels the same physical point as xµ (τ1 ) in some other coordinates. Therefore
hO1 (x0 (τ1 )) · · · On (x0 (τn ))i should be a physical prediction of the theory, which answers
the physical question ‘If I follow the path xµ0 (τ ), carrying an O-meter, what do I
measure?’. So apparently, to construct diff-invariant physical observables, we need to
tie them to infinity. Although this sounds like a straightforward fix, it is actually a
radical departure from ordinary, local quantum field theory.
The graviton is not composite (Weinberg-Witten theorem)
In QCD, there are quarks at high energies, and pions are composite degrees of freedom
that appear at low energy where the quarks are strongly coupled. The pion Lagrangian
is non-renormalizable; it breaks down at the QCD scale and must be replaced by the
full UV-complete theory of QCD.
Based on this analogy, we might guess that the UV completion of gravity is an ordinary,
D = 4 quantum field theory with no graviton, and that the graviton is a emergent
degree of freedom at low energies. This is wrong. The graviton may be an emergent
degree of freedom, but it cannot come from an ordinary D = 4 quantum field theory
in the UV. The reason is the Weinberg-Witten theorem:†
∗
This is true in the effective field theory of gravity too, not just in the UV. However in perturbation
theory it is not a problem. In perturbation theory, coordinates xµ label points on the fixed background
manifold, which are meaningful. It is only when we allow the geometry to fluctuate wildly that this
really becomes a problem.
†
See the original paper for the proof, it is short and clear. The basic idea is to first argue that,
since states carry energy, hp|T 00 (0)|pi 6= 0. However, if these are single-particle states for a particle
with helicity ±2, then there is just now way for hp0 |Tµν |pi to transform properly under rotations unless
it vanishes.
16
A 4D Lorentz-invariant QFT with a conserved, gauge-invariant stress tensor Tµν cannot
have massless particles with spin > 1.
This does not rule out general relativity itself because Tµν is not gauge invariant in GR
(or, equivalent, the physical part of Tµν is not really a Lorentz tensor). It does rule
out a composite graviton: If gravity emerged from an ordinary QFT, then in the UV
there is no diffeomorphism symmetry, the stress tensor is gauge invariant, so there can
be no graviton in the spectrum.
Emergent spacetime
So the theory of the graviton is sick in the UV, but if we stick to ordinary QFT we
cannot eliminate the graviton in the UV. This leaves two possibilities. One is that
the graviton appears in the UV theory, along with other degrees of freedom which
cure the problems seen in effective field theory. The other is that the graviton is an
emergent degree of freedom, but the UV theory is not an ordinary 4D QFT. These
are not mutually exclusive, and in fact both of these possibilities are realized in string
theory (simultaneously!).
In this course we will focus on the second possibility. We will discuss models where
not only the graviton, but spacetime itself is emergent. The fundamental degrees of
freedom of the theory do not live in the same spacetime as the final theory, or in
some cases do not live in any spacetime at all. Spacetime is an approximate, collective
description of these underlying degrees of freedom, and makes sense only the infrared.
The graviton is emergent, but evades Weinberg-Witten because the way it emerges is
outside the usual framework of QFT.
A look ahead
There are many ways to approach this subject. In this course we will take a route that
begins and ends with black holes. Unlike other EFTs (eg the pion Lagrangian), the
Einstein action contains an enormous amount of information about the UV completion
— infrared hints about the ultraviolet. Much of this information is encoded in the
thermodynamics of black holes, so that is our starting point, and will be the basis of
the first half of the course. As it turns out, black holes also lead to emergent spacetime
and the AdS/CFT correspondence, which are the topics of the second half of the course.
17
1.3
Homework
Review the chapter on black holes in Carroll’s textbook (or online lecture notes) on
General Relativity.
18
2
The Laws of Black Hole Thermodynamics
In classical GR, black holes obey ‘laws’ that look analogous to the laws of thermodynamics. These are classical laws that follow from the Eintsein equations. Eventually,
we will see that in quantum gravity, this is not just an analogy: these laws are the
ordinary laws of thermodynamics, governing the microsopic UV degrees of freedom
that make up black holes.
2.1
Quick review of the ordinary laws of thermodynamics
The first law of thermodynamics is conservation of energy,
∆E = Q
(2.1)
where Q is the heat transferred to the system.∗ For quasistatic (reversible) changes
from one equilibrium state to a nearby equilibrium state, δQ = T dS so the 1st law is
T dS = dE .
(2.2)
Often we will turn on a potential of some kind. For example, in the presence of an
ordinary electric potential Φ, the 1st law becomes
T dS = dE − ΦdQ
(2.3)
where Q is the total electric charge. If we also turn on an angular potential, then the
1st law is
T dS = dE − ΩdJ − ΦdQ .
(2.4)
The second law of thermodynamics is the statement that in any physical process,
entropy cannot decrease:
∆S ≥ 0 .
(2.5)
These laws can of course be derived (more or less) from statistical mechanics. In the
microscopic statistical theory, the laws of thermodynamics are not exact, but are an
∗
Often the rhs is written Q + W where −W is the work done by the system. We’ll set W = 0.
19
extremely good approximation in a system with many degrees of freedom.
2.2
The Reissner-Nordstrom Black Hole
The Riessner-Nordstrom solution is a charged black hole in asymptotically flat space.
It will serve as an example many times in this course.
Consider the Einstein-Maxwell action (setting units GN = 1),∗
1
S=
16π
Z
√
d4 x −g (R − Fµν F µν )
(2.6)
where Fµν = ∇µ Aν − ∇ν Aµ = ∂µ Aν − ∂ν Aµ . This describes gravity coupled to electromagnetism. The equations of motion derived from this action are
Rµν − 12 gµν R = 8πTµν
(2.7)
∇µ F µν = 0
(2.8)
with the Maxwell stress tensor
Tµν
1
2 δS matter
√
=
=−
µν
−g δg
4π
1
− gµν Fαβ F αβ + Fµγ Fν γ
4
.
(2.9)
The Reissner-Nordstrom solution is
ds2 = −f (r)dt2 +
with
f (r) = 1 −
dr2
+ r2 dΩ22
f (r)
Q2
2M
+ 2 ,
r
r
(2.10)
(2.11)
and an electromagnetic field
Q
Aµ dxµ = − dt,
r
so
Frt =
Q
.
r2
(2.12)
This component of the field strength is the electric field in the radial direction, so this
is exactly the gauge field corresponding to a point source of charge Q at r = 0.
∗
See Carroll Chapter 6 for background material.The factors of 2 in that chapter are confusing; see
appendix E of Wald for a consistent set of conventions similar to the ones we use here.
20
This is a static, spherically symmetric, charged black hole. There is nothing on the rhs
of the Maxwell equation (2.8), so the charge is carried by the black hole itself; there are
no charged particles anywhere. The parameter Q in the solution is the electric charge;
this can be verified by the Gauss law,
Qelectric
1
=
4π
Z
1 2
r
?F =
4π
∂Σ
Z
dΩ2 Frt = Q .
(2.13)
This integral is over the boundary of a fixed-time slice Σ, ie a surface of constant t and
constant r 1.
Horizons and global structure
Write
f (r) =
1
(r − r+ )(r − r− ),
r2
r± = M ±
p
M 2 − Q2 .
(2.14)
Then r+ is the event horizon and r− is the Cauchy horizon (also called the outer and
inner horizon). The coordinates (2.10) break down at the event horizon, though the
geometry and field strength are both smooth there. There is a curvature singularity at
r = 0. See Carroll’s textbook for a detailed discussion, and for the Penrose diagram of
this black hole.
We will always consider the case M > Q > 0. If |Q| > M , then r+ < 0, so the
curvature singularity is not hidden behind a horizon. This is called a naked singularity,
and there are two reasons we will ignore it: First, there is a great deal of evidence for
the cosmic censorship conjecture, which says that reasonable initial states never lead
to the creation of naked singularities.∗ Second, if there were a naked singularity, then
physics outside the black hole depends on the UV (since the naked singularity can spit
out visible very heavy particles), and we should not trust our effective theory anyway.
∗
There are also interesting violations of this conjecture in some situations, but in mild ways [Pretorious et al.]
21
2.3
The 1st law
Now we will check that this black hole obeys an equation analogous to (2.3), if we
define an ‘entropy’ proportional to the area of the black hole horizon:
S≡
1
× Area of horizon .
4~GN
(2.15)
We’ve temporarily restored the units in order to see that this is the area of the horizon
√
in units of the Planck length `P = ~GN . (Now we’ll again set GN = ~ = 1.) For now
this is just a definition but we will see later that there is a deep connection to actual
2
entropy. The horizon has metric ds2 = r+
dΩ22 , so the area is simply
2
A = 4πr+
= 4π(M +
p
M 2 − Q2 )2 .
(2.16)
Varying the entropy gives
dS =
4π
0
f (r+ )
dM −
4πQ
0
f (r+ )r+
dQ .
(2.17)
Rearranging, this can be written as the 1st law in the form
T dS = dM − ΦdQ
(2.18)
with
p
M 2 − Q2
r+ − r−
p
T ≡
=
,
2
4πr+
2π(M + M 2 − Q2 )2
Φ=
Q
Q
p
=
.
r+
M + M 2 − Q2
(2.19)
M , the mass of the black hole, is the total energy of this spacetime, so this makes
sense. Φ also has a natural interpretation:
Φ = −A0 |r=r+ .
(2.20)
It is the electric potential of the horizon.
But, we have no good reason yet to call T the ‘temperature’ or S the ‘entropy’ (this
will come later).
22
The 1st law relates two nearby equilibrium configurations. There are two ways we
can think about it: (i) as a mathematical relation on the space of solutions to the
equations, or (ii) dynamically, as what happens to the entropy if you throw some
energy and charge into the black hole.
T is related to the surface gravity of the black hole
T =
κ
,
2π
(2.21)
which is defined physically as the acceleration due to gravity near the horizon (which
goes to infinity) times the redshift factor (which goes to zero). If you stand far away
from the black hole holding a fishing pole, and dangle an object on your fishing line
near so it hovers near the horizon, then you will measure the tension in your fishing
line to be κMobject . It can be shown that κ is constant everywhere on the horizon
of a stationary black hole. This is analogous to the ‘0th law of thermodynamics’: in
equilibrium, temperature is constant.
If we restore units, then note that S ∝ ~, so
T ∝~.
(2.22)
Exercise: Thermodynamics of 3d Black Holes
Difficulty level: easy
Three-dimensional gravity has no true graviton, since a massless spin-2 particle has
1
D(D
2
− 3) = 0 local degrees of freedom. However, with a negative cosmological
constant, there are non-trivial black hole solutions, found by Banados, Teitelboim, and
Zanelli. The metric of the non-rotating BTZ black hole is
2
ds = `
2
dr2
−(r − 8M )dt + 2
+ r2 dφ2
r − 8M
2
2
,
(2.23)
where φ is an angular coordinate, φ ∼ φ + 2π. This is a black hole of mass M in a
spacetime with cosmological constant Λ = − `12 .
(a) Compute the area of the black hole horizon to find the entropy.
23
(b) Vary the entropy, and compare to the 1st law T dS = dM to find the temperature
of the black hole.
(c) Put all the factors of GN and ~ back into your formulas for S and T . S should
be dimensionless and T should have units of energy (since by T we always mean
T ≡ kB Tthermodynamic ). Does T have any dependence on GN ?
Exercise: Thermodynamics of rotating black holes.
Difficulty level: Straightforward, if you do the algebra on a computer
The Kerr metric is
ds2 = −
∆(r)
ρ2
1
2
2
(dt−a
sin
θdφ)
+
dr2 +ρ2 dθ2 + 2 sin2 θ(adt−(r2 +a2 )dφ)2 , (2.24)
2
ρ
∆(r)
ρ
where
∆(r) = r2 + a2 − 2M r ,
ρ2 = r2 + a2 cos2 θ ,
(2.25)
and −M < a < M . This describes a rotating black hole with mass M and angular
momentum
J = aM .
(2.26)
(a) Show that the entropy is
S = 2πM r+ = 2πM (M +
√
M 2 − a2 ) .
(2.27)
(b) The first law of thermodynamics, in a situation with an angular potential Ω, takes
the form
T dS = dM − ΩdJ .
(2.28)
Use this to find the temperature and angular potential of the Kerr black hole in terms
of M, a. (Hint: The angular potential can also be defined as the angular velocity of
tt
the horizon: Ω = − ggtφ
|r=r+ .)
24
2.4
The 2nd law
The second law of thermodynamics says that entropy cannot decrease: ∆S ≥ 0. This
law does not require a quasistatic process; it is true in any physical process, including
those that go far from equilibrium. (For example, if gas is confined to half a box, and
we remove the partition.)
Hawking proved, directly from the Einstein equation, that in any physical process
the area of the event horizon can never decrease. This parallels the second law of
thermodynamics! This is a very surprising feature of these complicated nonlinear
PDEs. We will not give the general proof; see Wald’s textbook.
Exercise: Black hole collision
Difficulty level: 2 lines
The 2nd law also applies to multiple black holes. In this case the statement is that the
total entropy – ie the sum of the areas of all black holes – must increase. Argue that if
two uncharged, non-rotating black holes collide violently to make one bigger black hole,
then at most 29% of their initial rest energy can be radiated in gravitational waves.
Exercise: Perturbative 2nd law
Difficulty level: Easy if familiar with particle motion on black holes
The 2nd law applies to the full nonlinear Einstein equation. In most cases, like a
black hole collision, it is hopeless to actually solve the Einstein equations explicitly
and check that it holds. But one special case where this can be done is for small
perturbations of a black hole. In this exercise we will drop a charged, massive particle
into a Reissner-Nordstrom black hole, and check that the entropy increases.∗
Suppose we drop a particle of energy and charge q into a Reissner-Nordstrom black
hole along a radial geodesic (to avoid adding angular momentum), with M and
∗
Reference: MTW section 33.8.
25
q Q. This will change the mass and charge of the black hole,
M → M + ,
Q→Q+q .
(2.29)
Although initially there will be some fluctuations in the spacetime and ripples on the
horizon from the particle that just passed through, these will quickly decay so that
we have once again the Reissner-Nordstrom solution, now with the new energy and
charge. Therefore, in this process the area of the black hole horizon changes according
to the 1st law (2.18),
δS =
1
( − Φq) .
T
(2.30)
(a) The infalling particle follows a trajectory xµ (τ ) where τ is proper time. Its 4momentum is
pµ =
dxµ
.
dτ
(2.31)
In a spacetime with a time-translation Killing vector ζ (t) , the energy of a charged
particle
= −(p + qA) · ζ (t) .
(2.32)
This is conserved along the path of the particle (which is not a geodesic, since it feels an
electromagnetic force). For a charged particle on the Reissner-Nordstrom black hole,
find in terms of f (r) and the components of pµ .
(b) Assume Q > 0. For one sign of q, the energy can be negative. Which sign?
If we drop a negative-energy particle into a black hole, the mass of the black hole
decreases. Therefore it is possible to extract energy using this process. For uncharged
but rotating black holes, a similar procedure can be used to extract energy in what
is called the Penrose process. Particles far from the black hole cannot have negative
energy, so negative-energy orbits are always confined to a region near the horizon. This
region is called the ergosphere.
(c) Although we can decrease the energy of the charged black hole, we cannot decrease
the entropy. To show this, we need to find the minimal energy of an orbit crossing the
horizon. Assume the particle enters the horizon along a purely radial orbit,∗ pθ = pφ =
∗
This assumption is not necessary. In the general case, the particle can add angular momentum to
the black hole, so we need to consider the charged, rotating Kerr-Newman spacetime. This is treated
26
0. The proper time along the orbit is
dτ 2 = −ds2 = f (r)dt2 −
dr2
.
f (r)
(2.33)
Use this equation to write in terms of pr , q, and f (r).
(d) is conserved along the orbit, so you can evaluate it where the particle crosses the
horizon, r = r+ . Show that the minimal value of is
min = qAt (r = r+ ) .
(2.34)
Conclude that the 2nd law of thermodynamics is obeyed.
(e) Reversible processes are those in which ∆S = 0. How would you reversibly drop
a charged particle into a Reissner-Nordstrom black hole? (i.e., what charge would it
have and how would you drop it?)
2.5
Higher curvature corrections
Everything in this section so far has assumed the gravity action is
R√
−g(R − 2Λ). As
discussed in section 1.1, this is incomplete: there should be higher curvature corrections
suppressed by the scale of new physics.
In a general theory of gravity including curvature corrections, the formula for the
entropy also receives corrections,
S=
Area
+ higher curvature corrections .
4
(2.35)
The more general formula is called the ‘Wald entropy’. We will postpone the general
discussion of the Wald entropy until later; for now suffice it to say that the 1st law still
holds. The 2nd law, however, does not. There are known counterexamples involving
black hole collisions.∗ To my knowledge this is not fully understood. A likely explanain detail in [MTW section 33].
∗
See arXiv: [hep-th/9305016], [0705.1518], [1011.4988].
27
tion is that this signals a breakdown of the effective field theory — i.e., that when these
violations occur we must include higher corrections or corrections from new physics in
the UV.
2.6
A look ahead
We have seen that the classical Einstein equations lead to laws of black hole mechanics
that are analogous to the laws of thermodynamics. In quantum gravity, it is not just
an analogy.
Temperature
What we called ‘T ’ is a true temperature: black holes radiate as blackbodies with
temperature T . This is Hawking radiation. It does not rely on quantizing gravity itself
— it is a feature of quantum field theory in curved space, which will be derived in the
next couple lectures.
Generalized second law
The entropy S is also a real entropy. This means that the total entropy of a system
is the ordinary entropy (of whatever gas is present, or a cup of tea, etc) plus the total
entropy of all the black holes in the system. The generalized second law is the statement
that the total entropy cannot decrease:
Stot = Sblack
holes
+ Sstuf f ,
∆Stot ≥ 0 .
(2.36)
If you throw a cup of hot tea into a black hole, then this entropy seems to vanish. This is
puzzling, because if we didn’t know about black hole entropy, we might conclude that
the ordinary 2nd law (applied to the tea) had been violated by destroying entropy.
However, the generalized second law guarantees that in this process the area of the
horizon will increase, and this will (at least) make up for lost entropy of the tea.
Counting microstates
Finally, we know that in quantum mechanics, entropy is supposed to count the states
28
of a system:
S(E) = log (# states with energy E) .
(2.37)
For a supermassive astrophysical black hole like
the one at the center of the Milky Way,
this is an enormous number, or order
106 km
`P
2
∼ 1088 . For comparison, the entropy
of all baryons in the observable universe is around 1082 , and the entropy of the CMB
is about 1089 . So black holes must have enormous number of states!
Classical black holes have no microstates. They are completely specified by M, J, Q
(this statement is called the no hair theorem). How, then, can they have entropy? The
answer should be that in the UV completion of quantum gravity, black holes have many
microstates. This is exactly what happens in certain examples in string theory, and in
AdS/CFT, as we’ll see later. Unlike Hawking radiation, to understand the microscopic
origin of black hole entropy requires the UV completion of quantum gravity. Turning
this around, this means that black hole entropy is a rare and important gift from
nature: an infrared constraint on the ultraviolet completion, that we should take very
seriously in trying to quantize gravity.
29
3
Rindler Space and Hawking Radiation
The next couple of lectures are on Hawking radiation. There are many good references to learn this subject, for example: Carroll’s GR book Chapter 9; Townsend
gr-qc/9707012; Jacobson gr-qc/0308048. Therefore in these notes I will go quickly
through some of the standard material, but slower through the material that is hard
to find elsewhere. I strongly recommend reading Carroll’s chapter too.
Hawking radiation is a feature of QFT in curved spacetime. It does not require that
we quantize gravity – it just requires that we quantize the perturbative fields on the
black hole background. In fact we can see very similar physics in flat spacetime.
3.1
Rindler space
2d Rindler space is a patch of Minkowski space. In 2D, the metric is
ds2 = dR2 − R2 dη 2 .
(3.1)
There is a horizon at R = 0 so these coordinates are good for R > 0, η =anything.
Notice the similarity to polar coordinates on R2 , if we take η → iφ. This suggests
the following coordinate change from ‘polar-like’ coordinates to ‘Cartesian-like’ coordinates,
x = R cosh η,
t = R sinh η .
(3.2)
The new metric is just Minkowski space R1,1 ,
ds2 = −dt2 + dx2 .
(3.3)
Looking at (3.2), we see x2 − t2 = R2 > 0, so the Rindler coordinates only cover the
patch of Minkowski space with
x > 0,
|t| < x .
This is the ‘right wedge’, which covers one quarter of the Penrose diagram.
30
(3.4)
Higher dimensional Rindler space is
ds2 = dR2 − R2 dη 2 + d~y 2 ,
(3.5)
so we can map this to a patch of R1,D−1 by the same coordinate change. The other
coordinates just come along for the ride.
A ‘Rindler observer’ is an observer sitting at fixed R. This is not a geodesic — it is a
uniformly accelerating trajectory. You can check this by mapping back to Minkowski
space. Rindler observers are effectively ‘confined’ to a piece of Minkowski space, and
they see a horizon at R = 0. This horizon is in many ways very similar to a black hole
horizon.
Exercise: Rindler time translations are Minkowski boosts
Difficulty level: a few lines
ζ(η) = ∂η is an obvious Killing vector of Rindler space, since the metric is independent
of η. By explicitly transforming this vector to Minkowski coordinates, show that it is
a Lorentz boost.∗
3.2
Near the black hole horizon
Black holes have an approximate Rindler region near the horizon. For example, start
with the Schwarzschild solution
ds2 = −f (r)dt2 +
dr2
+ r2 dΩ22 ,
f (r)
f (r) = 1 −
2M
.
r
(3.6)
Make the coordinate change
r = 2M (1 + 2 ),
∗
so f (r) ≈ 2
Reminder: The notation ζ = ∂η means, in components, ζ µ ∂µ = ∂η , i.e., ζ = η̂.
31
(3.7)
and expand the metric at small ,
ds2 = −2 dt2 + 16M 2 d2 + 4M 2 dΩ22 + · · ·
(3.8)
The (t, ) piece of this metric is Rindler space (we can rescale t and to make it look
exactly like (3.1)).
Although 2d Rindler is a solution of the Einstein equations, the metric written in (3.8)
(excluding the dots) is R1,1 × S 2 . This is not a solution of the Einstein equations. It
is only an approximate solution for small .
3.3
Periodicity trick for Hawking Temperature
Now we will give a ‘trick’ to derive the Hawking temperature. The trick is to argue that
that, in the black hole metric, the time coordinate must be periodic in the imaginary
direction, and this imaginary periodicity implies that the black hole has a temperature.
Actually, this trick is completely correct, and we will justify it later from the path
integral, but don’t expect this subsection to be very convincing yet!
First we want to argue that QFT at finite temperature is periodic in imaginary time,
with periodicity
t ∼ t + iβ,
β = 1/T .
(3.9)
We will return this in detail later, but for now one way to see it is by looking at the
thermal Green’s function∗
Gβ (τ, x) ≡ − Tr ρthermal TE [O(τ, x)O(0, 0)] = −
1
Tr e−βH TE [O(τ, x)O(0, 0)] , (3.10)
Z
where τ = it is Euclidean time, and TE means Euclidean-time ordering (i.e., put the
∗
This definition holds for −β < τ < β. A good reference for the many types of thermal Green’s
functions is Fetter and Walecka, Quantum Theory of Many-particle Systems, 1971, in particular chapters 7 and 9. The function Gβ defined here is equal to the Euclidean Green’s function on a cylinder
that we will discuss later (up to normalization).
32
larger value of τ on the left). This is periodic in imaginary time,∗
1
Tr e−βH O(τ, x)O(0, 0)
Z
1
= − Tr O(0, 0)e−βH O(τ, x)
Z
1
= − Tr e−βH O(β, 0)O(τ, x)
Z
= Gβ (τ − β, x)
Gβ (τ, x) = −
(3.11)
(3.12)
(3.13)
(3.14)
Now returning to black holes, Rindler space (3.1) is related to polar coordinates, dR2 +
R2 dφ2 with η = iφ. Polar coordinates on R2 are singular at the origin unless φ is a
periodic variable, φ ∼ φ + 2π. Therefore η is periodic in the imaginary direction,
η ∼ η + 2πi .
(3.15)
Going back through all the coordinate transformations relating the near horizon black
hole to Rindler space, this implies that the Schwarzschild coordinate t has an imaginary
periodicity
t ∼ t + iβ ,
β ≡ 8πM .
(3.16)
Comparing to finite temperature QFT,
T =
1
1
=
.
β
8πM
(3.17)
This agrees with the Hawking temperature derived from the first law, (2.19) (setting
Q = 0). As mentioned above, this derivation probably is not very convincing yet, but
it is often the easiest way to calculate T given a black hole metric.
Exercise: Schwarzschild periodicity
Difficulty level: easy
Work through the coordinate transformations in section (3.2) to relate the Schwarzschild
near-horizon to Rindler space, and show that (3.15) implies (3.16).
∗
1st line: definition (we assume 0 < τ < β, so this is τ -ordered). 2nd line: cyclicity of trace. 3rd
line: definition of time translation, O(τ, x) = eτ H O(0, x)e−τ H . 4th line: definition.
33
Exercise: Kerr periodicity
Difficulty level: moderate – a few pages
Field theory at finite temperature and and angular potential is periodic in imaginary
time, but with an extra shift in the angular direction:
(t, φ) ∼ (t + iβ, φ − iβΩ) .
(3.18)
(a) Derive (3.18) by an argument similar to (3.11). The density matrix for QFT at
finite temperature and angular potential is ρ = e−β(H−ΩJ) where J is the angular
momentum.
(b) Starting from the Kerr metric (11.22), we will follow steps similar to section 3.2
to relate the near horizon to Rindler space. This will be easiest if you write ∆(r) =
(r − r+ )(r − r− ) and work in terms of r± instead of plugging in all the M ’s and a’s.
Also, you can safely ignore that θ-direction by setting θ = π/2 (you should come back
at the end of the problem and convince yourself that this was reasonable).
Plug in r = r+ (1 + 2 ) and expanding in . You should find something of the form
ad2 + (bdφ − cdt)2 + 2 (edφ + f dt + · · · ) + · · · .
(3.19)
where a, b, c, e, f are some constants.
(c) To find the correct periodicity in this situation, define the corotating angular coordinate φ̃ = bφ − ct (this coordinate rotates with the horizon). Now the usual Rindler
argument implies that t has an imaginary identification with φ̃ held fixed. Translate
this identification back into t, φ coordinates and compare to (3.18) to read off β and
Ω. Check that your answers agree with the ones you derived from the first law in the
exercise around (11.22).
34
3.4
Unruh radiation
Suppose our spacetime is Minkowski space, in the vacuum state. An observer on a
worldline of fixed x will not observe any excitations. So, if the observer carries a
thermometer, then the thermometer will read ‘temperature = 0’ for all time. More
generally, if the observer carries any device with internal energy levels that can be excited by interacting with whatever matter fields exist in the theory (an Unruh detector ),
then this device will forever remain in its ground state.
However, now consider a uniformly accelerating observer with acceleration a. This
corresponds to a Rindler observer sitting at fixed R = 1/a. We will show that in
the same quantum state – the Minkowski vacuum – this observer feels a heat bath at
temperature T =
a
.
2π
No unique vacuum
Thus the Minkowski vacuum is not the same as the Rindler vacuum. This is a general
feature of quantum field theory in curved space (although in this example spacetime
is flat!). In general, there is no such thing as the vacuum state, only the vacuum state
according to some particular observer. The reason for this is the following. In QFT,
we expand quantum fields in energy modes,
φ̂ =
X aω,k eiωt−ikx + a†ω,k e−iωt+ikx
.
(3.20)
ω>0,k
The ‘vacuum’ is defined as the state annihilated by the negative energy modes:
aω |0i = 0 ,
(3.21)
and excitations are created by the positive-energy modes a† .
The ambiguity comes from the fact that energy is observer dependent. The energy is
the expectation value of the Hamiltonian; and the Hamiltonian is the operator that
generates time evolution
i
[H, O] = ∂t O .
~
(3.22)
Therefore the Hamiltonian depends on a choice of time t. In GR, we are free to call
35
Figure 1: Different choices of time in Minkowski vs Rindler space.
any timelike direction t. Different choices of this coordinate correspond to different
choices of Hamiltonian, and therefore different notions of positive energy, and therefore
different notions of vacuum state. See figure.
The Unruh temperature
The quick way to the Unruh temperature is the imaginary-time periodicity trick. Since
the Rindler time coordinate η is periodic under η ∼ η+2πi, this looks like a temperature
T =
1
.
2π
This is the temperature associated to the time translation ∂η . In other words,
if Hη is the Hamiltonian that generates η-translations, then the periodicity trick tells
us that the density matrix for fields in Rindler space is
ρRindler = e−2πHη .
(3.23)
The proper time of a Rindler observer at fixed R = R0 is
dτ = R0 dη .
(3.24)
Therefore R0 is the redshift factor, and the temperature actually observed (say by a
thermometer) is
1 1
1
a
Tthermometer = √
=
=
.
gηη 2π
2πR0
2π
(3.25)
Where in this argument did we actually decide which state we are in? There are
excitations of Rindler/Minkowski space – say, a herd of elephants running by – where
an observer will certainly not measure a uniform heat bath of thermal radiation! The
answer is that by applying the periodicity trick, we have actually selected one particular
36
very special state, (3.23). What we still need to show is that this state is in fact the
Minkowski vacuum.
First, some FAQ:∗
• Does a Minkowski observer see that the Rindler observer is detecting particles?
Yes, the Minkowski observer can see that the Rindler observer’s particle detector
is clicking. How does that make any sense if the Minkowski observer doesn’t see
the particles being absorbed? The Minkowski observer actually sees the Rindler
observer’s detector emit a particle when it clicks. This can be worked out in
detail, but the easy way to see this is that the absorption of a Rindler mode
changes the quantum state of the fields; the only way to change the vacuum
state is to excite something.
• Doesn’t this violate conservation of energy? No, the Rindler observer is uniformly
accelerating. So this observer must be carrying a rocket booster. From the point
of view of a Minkowski observer, the rocket is providing the energy that excites
the Rindler observer’s thermometer, and causes Minkowski-particle emission.
• How hot is Unruh radiation? Not very hot. In Kelvin, T = ~a/(2πckB ). To
reach 1K, you need to accelerate at 2.5 × 1020 m/s2 .
• According to the equivalence principle, sitting still in a uniform gravitational field
is the same as accelerating. So do observers sitting near a massive object see
Unruh radiation?
Only if it’s a black hole (as discussed below). Observers
sitting on Earth do not see Unruh radiation because the quantum fields near the
Earth are not in the state (3.23). This is similar to the answer to the question
‘why doesn’t an electron sitting on the Earth’s surface radiate?’.
Back to Minkowski
One very explicit method to show that the thermal Rindler state is in fact the Minkowski
vacuum is to compare the Rindler modes to the Minkowski modes, and check that iminkowski
posing aM
|0i = 0 leads to the state (3.23) in the Rindler half-space. This is
w
already nicely written in Carroll’s GR book and many other places so I will not bother
∗
See http://www.scholarpedia.org/article/Unruh effect for further discussion and references.
37
writing it here. Read the calculation there. But don’t be fooled – that method gives
the impression that the Rindler temperature has something to do with the modes of a
free field in Minkowski space. This is not the case. The Rindler temperature is fixed by
symmetries, and holds even for strongly interacting field theories, for example QCD.
The general argument will be given in detail below, using Euclidean path integrals.
Exercise: Rindler Modes
Work through the details of section 9.5 in Carroll’s GR textbook Spacetime and Geometry.
3.5
Hawking radiation
We have shown that Unruh observers see a heat bath set by the periodicity in imaginary
time. We have also seen that the near-horizon region of a black hole is Rindler space.
Putting these two facts together we get Hawking radiation: black holes radiate like a
blackbody at temperature T . Some comments are in order:
1. This section has argued intuitively for Hawking radiation but don’t be disturbed
if you find the argument unconvincing. There are two ways to give a more
explicit and more convincing derivation. One is to match in modes to out modes
and calculate Bogoliubov coefficients; I recommend you read this calculation in
Carroll’s book. The other method is using Euclidean path integrals to put the
imaginary-time trick on a solid footing. We will follow this latter method in the
next section.
2. This is the same T that appeared in the 1st law of thermodynamics. Historically,
the 1st and 2nd law were discovered before Hawking radiation. Since T and S
show up together in the 1st law, it was only possible to fix each of them up to
a proportionality factor. This ambiguity was removed when Hawking discovered
(to everyone’s surprise) that black holes actually radiate.
38
3. In this derivation of Hawking radiation we have chosen a particular state by
applying the imaginary-time periodicity trick. In the Unruh discussion, this trick
selected the Minkowski vacuum. Another way to say it is that the imaginarytime trick picks a state in which the stress tensor is regular on the past and
future Rindler horizon. Therefore, by applying this trick to black holes, we have
selected a particular quantum state which is regular on the past and future event
horizon. This state is called the Hartle-Hawking vacuum. Physically, it should be
interpreted as a black hole in thermodynamic equilibrium with its surroundings.
Others commonly discussed are:
(a) The Boulware vacuum is a state with no radiation. In Rindler space, it
corresponds to the Rindler vacuum, ρRindler = |0ih0|. This state is singular on
the past and future Rindler/event horizons, so is not usually physically relevant.
(b) The Unruh vacuum is a state in which the black hole radiates at temperature
T , but the surroundings have zero temperature. This is the state of an astrophysical black hole formed by gravitational collapse. It is regular on the future
event horizon, but singular on the past even horizon (which is OK because black
holes formed by collapse do not have a past event horizon). If we put reflecting
boundary conditions far from the black hole to confine the radiation in a box, or
if we work in asymptotically anti-de Sitter space, then the Unruh state eventually
equilibrates so at late times is identical to the Hartle-Hawking state.
4. If you stand far from a black hole, you will actually not quite see a blackbody. A
Rindler observer sees an exact blackbody spectrum, as does an observer hovering
near a black hole horizon. But far from the black hole, the spectrum is modified
by a ‘greybody factor’ which accounts for absorption and re-emission of radiation
by the intervening geometry. Far from a black hole, the occupation number of a
mode with frequency ω is
hnω i =
1
× σabs (ω) .
−1
eβω
(3.26)
The first term is the blackbody formula and the second term is the greybody
factor. The greybody factor is equal to the absorption cross-section of a mode
with frequency ω hitting the black hole, since transmission into the black hole is
equal to transmission out of the black hole.
39
5. In Rindler space, an observer on a geodesic (i.e., Minkowski observer) falls
through the Rindler horizon, and this observer does not see the Unruh radiation. Similarly, you might expect that freely falling observers jumping into a
black hole will not see Hawking radiation. This is almost correct, as long as the
infalling observer is near the horizon in the approximately-Rindler region, but
not entirely — the potential barrier between the horizon and r = ∞ causes some
of the radiation to bounce back into the black hole, and this can be visible to an
infalling observer.
40
4
Path integrals, states, and operators in QFT
To put our derivation of Hawking radiation on a solid footing, and for other applications
to gravity later on, we will now take a slight detour to explain the relationship between
path integrals and states in quantum field theory. (This is material not normally
covered in detail in QFT courses or books; it is assumed that the reader is already
familiar with path integrals at the level of Peskin and Schröder.)
4.1
Transition amplitudes
Path integrals define transition amplitudes. A Euclidean path integral defines a transition amplitude under evolution by e−βH :
−βH
hφ2 |e
Z
φ(τ =β)=φ2
|φ1 i =
Dφ e−SE [φ] .
(4.1)
φ(τ =0)=φ1
This involves a split into space and (Euclidean) time; φ1,2 is a boundary condition that
specifies data at a fixed time. Exactly what this path integral means depends on the
topology of space. If space is a plane (or line in 2d), then we depict this by
hφ2 |e−βH |φ1 i =
(4.2)
meaning it’s a Euclidean path integral over an infinite strip Rd−1 ×interval, with the
boundary conditions shown and the interval has length β.
41
If space is a sphere (or circle in 2d), then the appropriate path integral is
hφ2 |e−βH |φ1 i =
(4.3)
ie it is a path integral over a cylinder S d−1 ×interval, of length β.
4.2
Wavefunctions
The transition amplitude defines the wavefunction, in the Schroedinger picture. For
example the wavefunction for the state
|Ψi = |φ1 (τ )i = e−τ H |φ1 i
(4.4)
Ψ[φ2 ] ≡ hφ2 |Ψi .
(4.5)
is the overlap
4.3
Cutting the path integral
A ‘cut’ is a spatial slice of the Euclidean manifold. It is a codimension-1 surface Σ.
To define the transition amplitude, we specified data on two cuts, at τ = 0 and τ = β.
We can formally think of a path integral with one set of boundary conditions and one
open cut as a quantum state. That is, the state
|Ψi = e−βH |φ1 i
42
(4.6)
is the path integral
Z
φ(τ =β)=??
|Ψi =
Dφ eS[φ] =
(4.7)
φ(τ =0)=φ1
This is a formal object where the data on the top cut is left unspecified. It is a
functional |Ψi that turns field data hφ2 | into complex numbers hφ2 |Ψi.
More generally, any path integral with an open cut Σ defines a quantum state on Σ.
For example, this Euclidean path integral in a 2D QFT defines some particular state
on a circle, Σ = S 1 :
|Xi =
(4.8)
The wavefunction of this state is computed by the path integral
X[φ2 ] = hφ2 |Xi =
(4.9)
43
We could also insert some operators into this path integral to get a different state:
|X 0 i =
(4.10)
This means a Euclidedan path integral weighted by O1 (x1 )O2 (x2 )e−SE [φ] , instead of
just the usual e−SE [φ] .
4.4
Euclidean vs. Lorentzian
So far we have discussed Euclidean path integrals. But states are states: they are
defined on a spatial surface and do not care about Lorentzian vs Euclidean. The state
|Xi, defined above by a Euclidean path integral, is a state in the Hilbert space of the
Lorentzian theory. It is defined at a particular Lorentzian time, call it t = 0. It can be
evolved forward in Lorentzian time by acting with e−iHt , or equivalently by performing
the Lorentzian path integral:
|X(t)i = e−iHt |Xi =
(4.11)
44
Since |Xi ≡ |X(0)i was defined by a Euclidean path integral, the state |X(t)i is a path
integral that is part Euclidean, part Lorentzian:
|X(t)i =
(4.12)
Again, this equation should be read as a formal definition of the state that tells you
what path integral to perform to compute transition amplitudes:
hφ2 |X(t)i =
4.5
(4.13)
The ground state
Evolution in Euclidean time damps excitations. Suppose we start in some state |Y i
and expand in energy eigenstates:
|Y i =
X
yn |ni,
H|ni = En |ni .
(4.14)
n
Then by evolving over a long Euclidean time we can project onto the lowest energy
state,
e−τ H |Y i ≈ e−τ E0 y0 |0i .
(τ → ∞)
(4.15)
It follows that we can define the (unnormalized) ground state by doing a path integral
that extends all the way to infinity in one direction. For example the ground state on
45
the line is produced by the Euclidean path integral
|0iline =
(4.16)
This means a path integral on the semi-infinite plane, with an open cut at the edge. The
ground state on a circle is produced by the path integral on a semi-infinite Euclidean
cylinder,
|0icircle =
(4.17)
These states are unnormalized.
4.6
Vacuum correlation functions
Path integrals with cuts can be glued together to make transition amplitudes. For
example, for a theory on a line, the vacuum-to-vacuum amplitude is
Z
h0|0i =
Dφe−SE [φ] =
(4.18)
The lower half-plane produces |0i, the upper half-plane produces h0|, and glueing them
together along the cuts at τ = 0 produces the transition amplitude. One way to see
that we should glue is to insert the identity:
h0|0i =
X
h0|φ1 ihφ1 |0i .
φ1
46
(4.19)
The first term is a path integral on the upper half plane; the second term is a path
integral on the lower half plane; and summing over all possible boundary conditions
φ1 in the middle just says that fields should be continuous across τ = 0 and therefore
glues the half-planes together.
Expectation values of local operators are computed by similar path integrals, but with
extra operator insertions. For example, correlation functions are expectation values in
the vacuum state. In Euclidean signature these are computed by the path integral
hO1 (x1 )O2 (x2 )i ≡ h0|O2 (x2 )O1 (x1 )|0i
=
(4.20)
(4.21)
This picture means the path integral of O1 (x1 )O2 (x2 )e−SE [φ] over fields on Rd . (The
ordering of operators does not matter on the lhs of (4.20), but is important on the rhs;
more in this below.)
Time-ordered Lorentzian vacuum correlation functions are computed by a more complicated, ‘folded’ path integral that is part Euclidean and part Lorentzian. For example
(assuming t1 > t2 ),
hO1 (t1 , ~x1 )O2 (t2 , ~x2 ) · · · i = h0| eiHt1 O1 (0, ~x1 )e−iHt1
47
eiHt2 O2 (0, ~x2 )e−iHt2 |0i (4.22)
is computed by the following path integral:
(4.23)
This path integral starts at t = −i∞ on the left; evolves to t = 0 to prepare the
vacuum state; evolves in Lorentzian time to t = t2 , where O2 is inserted; then evolves
to t1 where O1 is inserted; then evolves backwards in Lorentzian time to t = 0; then
evolves to t = +i∞ for the vacuum ‘bra’. Again, this picture means you should do the
path integral
Z
Dφ O1 (t1 , ~x1 )O2 (t2 , ~x2 )eiS[φ]
(4.24)
where we integrate over all fields φ defined on the mixed-signature manifold in the
picture. The Lorentzian action appears in this expression; when you integrate over the
Euclidean part of the manifold, the fact that t is imaginary will automatically change
this into e−SE [φ] .
We rarely need to think about doing folded path integrals like (4.23). Instead, we
do one of two equivalent things: (1) We compute the Euclidean path integral with
arbitrary values of the insertion points, then analytically continue to Lorentzian time,
48
or (2) We use an i prescription to compute the Lorentzian path integral. Actually the
usual i prescription is just a deformation of the integration contour (that is, integration
contour in field space) shown in figure (4.23), and computes exactly the same quantity.
So if you’re ever wondered what you were doing with that i, the answer is the figure
in (4.23)!
Aside: Some comments on time ordering
In a sense, time ordering does not really exist in Euclidean signature: fields commute,
h· · · O1 (x1 )O2 (x2 ) · · · i = h· · · O2 (x2 )O1 (x1 ) · · · i .
(4.25)
One way to see this is to note that correlators computed by the path integral
Z
Dφ O1 (x1 )O2 (x2 ) · · · e−SE
(4.26)
are just statistical averages, so they commute just like observables in stat-mech. Put
differently, the reason that fields don’t commute in Lorentzian signature is because
the correlator is not an analytic function of the coordinates. It has branch cuts when
O2 hits the light-cone of O1 , and requires an i prescription to define the function.
Different choices of i prescription give different types of correlation functions, and we
denote these different choices by writing the fields in a different order. In Euclidean
signature, correlators are analytic, there are not branch cuts, and there are no i’s, so
we don’t have to worry about how fields are ordered.
However, when we cut the path integral to translate to operator language, the field
operators don’t commute, even in Euclidean signature. They are ‘time’-ordered according to whatever slicing we choose for the path integral. So if states are defined
on constant-Euclidean-time slices, the path integral translates into an operator expression with fields ordered according to their Euclidean time. If states are defined on
constant-r slices (as we often do in conformal field theory), then the corresponding
operator expression has radially-ordered fields.
49
4.7
Density matrices
A density matrix is an operator; it takes a bra and a ket, and produces a complex
number. Thus any path integral with two open cuts defines a density matrix (unnormalized). For example, the density matrix ρ = e−βH , for a theory on a circle, is
formally the doubly-cut Euclidean path integral
ρ ≡ e−βH =
.
(4.27)
This is just a picture representing the statement that matrix elements hφ2 |ρ|φ1 i are
computed by the path integral with boundary conditions φ1,2 on the cuts.
4.8
Thermal partition function
The density matrix ρ = e−βH is the density matrix in a thermal ensemble at temperature T = 1/β. The thermal partition function is
Z(β) = tr e−βH .
(4.28)
This can be represented by a Euclidean path integral as follows:
Z(β) = tr e−βH
X
hφ1 |e−βH |φ1 i
=
(4.29)
(4.30)
φ1
=
X
(4.31)
φ1
50
In the last line, by summing over φ1 we are really just imposing periodic boundary
conditions on the cylinder. This glues together the two ends of the cylinder, producing
a torus. So the thermal partition function for a 2D theory on a circle is equal to a path
integral on a torus:
on circle: Z(β) =
(4.32)
Similarly, the thermal partition function for a 2D theory on a line is computed by a
path integral on an infinitely long cylinder of period β:
(4.33)
on line: Z(β) =
The trace ‘glues together’ parts of the Euclidean manifold that computes ρ.
The same thing works in higher-dimensional QFT at finite temperature: If space is a
plane Rd−1 , the thermal partition function is the path integral on Rd−1 × S 1 , and for
a theory on S d−1 , the thermal partition function is a path integral on S d−1 × S 1 .
4.9
Thermal correlators
Equal-time correlators at finite temperature are defined (up to normalization) by
hO1 (t = 0, ~x1 )O2 (t = 0, ~x2 ) · · · iβ ≡ Tr e−βH O1 (0, ~x1 )O2 (0, ~x2 ) · · ·
(4.34)
By the same logic, this is computed by a path integral on a cylinder Rd−1 × S 1 (if space
is a plane) or on S d−1 × S 1 (if space is a sphere).
51
To compute different-time Lorentzian correlators at finite temperature, the easist method
is usually to compute the Euclidean correlators first, as functions of arbitrary insertion
points on the Euclidean cylinder, then analyatically continue.
Finite-temperature correlators in 2d CFT
Difficulty level: moderate, a couple pages
In a 2d conformal field theory, the 2pt function on the Euclidean cylinder of size β is
fixed entirely by conformal invariance. Let w = x + itE be a complex coordinate on
the Euclidean cylinder (tE is Euclidean time and x is space). Then the 2pt function
on the cylinder is
hO(w1 , w̄1 )O(w2 , w̄2 )iβ =
1
sinh(2π(w1 − w2 )/β) sinh(2π(w̄1 − w̄2 )/β)
∆
(4.35)
where ∆ is called the scaling dimension of the operator O.
(a) Draw a picture of the path integral on the cylinder that computes (4.35).
(b) Translate your picture into operator language. Compare to (3.10). (Don’t worry
about the overall sign, this is a convention.)
(c) Check that the 2pt function written in (4.35) indeed has the periodicity of a thermal
correlator (see discussion around (3.10)).
(d) Analytically continue to find the finite-temperature 2pt function at real (Lorentzian)
times hO(t1 , x1 )O(t2 , x2 )i, where t is Lorentzian time. Don’t worry about which Lorentzian
ordering you are computing, just pick one. (The most obvious continuation will compute the time-ordered Lorentzian correlator.)
(e) Fix t1 = x1 = 0. Draw a picture of the complex-t2 plane showing the singularities
of (4.35). When you analytically continued in part (d), you implicitly chose a contour
in this plane to define the analytic continuation. Check that if (t2 , x2 ) lies inside the
future light-cone of (t1 , x1 ), then the analytic continuation is ambiguous, due to one of
the poles in the complex-t2 plane. This ambiguity is why timelike separated fields in
Lorentzian signature do not commute.
52
53
5
5.1
Path integral approach to Hawking radiation
Rindler Space and Reduced Density Matrices
We will use the Euclidean path integral to justify the claim in (3.23) that the Minkowski
vacuum corresponds to the Rindler state ρRindler = e−2πHη . Consider a 2d QFT on a
line, and let the state of the full system by the Minkowski vacuum,
ρ = |0ih0| .
(5.1)
As argued above, this state is prepared by a path integral on a half-plane, cut on the
line t = 0. Let us divide the line into x > 0 (region A) and x < 0 (region B). The
reduced density matrix in region A is
ρA ≡ trB ρ .
(5.2)
This has the nice property that all observables restricted to region A (or to the Rindler
wedge that is the causal evolution of region A) can be computed from ρA alone:
Tr ρO(x1 ) · · · O(xn ) = Tr ρA O(x1 ) · · · O(xn ) ,
for xi > 0, |t| < xi .
(5.3)
The path integral representation of ρA is
hφ2 |ρA |φ1 i =
X
hφ̃, φ2 |0ih0|φ1 , φ̃i
(5.4)
φ̃
=
(5.5)
The upper half of this diagram corresponds to the transition amplitude
P
φ̃ hφ̃, φ2 |0i
and the lower half to the transition amplitude h0|φ1 , φ̃i. The trace sums over fields in
54
the left Rindler wedge, which glues together these slits in the path integral, so in fact
hφ2 |ρA |φ1 i =
(5.6)
Now comes the key observation: we can re-slice this path integral by going to polar
coordinates dR2 + R2 dφ2 , and calling φ ‘time’. Let HRindler be the operator that
generates φ-evolution. That is,
1
[H, O] = ∂φ O
~
(5.7)
for any operator O. Then we can translate this same path integral back into operator language in a different way. That is, the path integral in (5.6) is equal to
hφ2 |e−2πHRindler |φ1 i. Therefore
ρA = e−2πHRindler .
(5.8)
This looks just like a thermal state at temperature 1/2π, but it is thermal with respect
to the rotation generator. When we go back to Minkowski space φ → iη, this becomes
the boost generator corresponding to the causal development of the Rindler wedge.
Therefore HRindler is exactly what we called Hη above.
This is a complete path-integral derivation of the statement that the Minkowski vacuum
leads to a thermal state in Rindler space. As mentioned above, this can also be shown
by explicit comparison of modes, but the path integral derivation can be more useful
for intuition. Another big advantage is that in the path integral derivation, we did not
assume anywhere that the matter fields were free, or even necessarily weakly coupled—
it is completely general.
Modular Hamiltonian
The Hamiltonian that appears in the relation ρRindler = e−2πHRindler is a special case
of a modular Hamiltonian. A modular Hamiltonian is simply defined as the log of a
density matrix (up to normalization). It is very useful for characterizing entanglement,
55
both in quantum gravity and in condensed matter physics.
5.2
Example: Free fields
So far, we have answered the question “What is the quantum state of fields on Rindler
space?” The complete answer is equation (5.8), and does not require any mention of
“particles” (which only make sense at weak coupling), or any particular observer.
However to gain a more concrete intuition for the physics it is very useful to think in
terms of particles. So in this subsection we will apply to result (5.8) to free (or weakly
interacting) fields, and discuss what an accelerating observer capable of detecting these
particles would actually experience.
A massless free field in 2D Rindler space (in Lorentz signature) obeys the wave equation
Φ = ∇µ ∇µ Φ = 0 .
(5.9)
Since η is our ‘time’ coordinate, we take the ansatz Φ = e−iωη f (R), and find the
solution
Φ = e−iωη+ik log R ,
ω2 = k2 ,
ω>0.
(5.10)
As usual in QFT, we expand the field operator in terms of creation and annihilation
operators,
Z
Φ̂(η, R) =
dk bk Φk + b†k Φ∗k
(5.11)
The creation operators b† create positive-energy modes φk . The b’s annihilate positiveenergy modes. The Rindler vacuum state is defined by
|0iR = bk |0iR = 0 ,
∀k .
(5.12)
It is clear that this is not the Minkowski vacuum state: Minkowski modes are expanded
in Minkowski plane waves, and Minkowski creation and annihilation operators a†k , ak
are not the same as the Rindler ones. The fact that Rindler space has a different choice
of ‘time’ means it has a different choice of ‘energy’ and therefore a different notion of
56
‘particle’ and ’vacuum’:
time coordinate ↔ energy ↔ particle ↔ vacuum .
(5.13)
How does this relate to our more abstract path integral discussion above? ω in the
mode-expansion (5.11) is exactly the eigenvalue of the Rindler Hamiltonian HRindler .
That is, the 1-Rindler-particle state,
|kiR = b†k |0iR
(5.14)
HRindler |kiR = ω|ki .
(5.15)
satisfies∗
Just like in flat space there are also multiparticle states.
In the Minkowski vacuum, the quantum state of these fields is ρRindler = e−2πHRindler .
We can use this to calculate observables. For example, what is the occupation number
of a mode with Rindler energy ω = |k|? The calculation is identical to the usual
blackbody calculation:
hnk i =
1
Tr e−2πHRindler b†k bk ,
Z
Z ≡ Tr e−2πHRindler
(5.16)
The number operator b†k bk counts the number of quanta in the mode, so it ranges from
n = 0 . . . ∞, so
!
hnk i =
X
ne−2πn|k| /
n≥0
=
1
e2π|k| − 1
!
X
e−2πn|k|
n≥0
.
(5.17)
This is of course the Planck blackbody spectrum.
What does an observer actually see?
An observer who has can detect the Φ-particle will see the blackbody spectrum (5.17).
However there is one last subtlety: an observer carrying a thermometer, or a calorimeter
∗
For the same reason that, in flat spacetime, if we write modes ∝ e−iωt then ω is the energy.
57
that measures energy in, say, joules, does not measure the energy ω. In fact, ω is
dimensionless, since the Rindler time coordinate is dimensionless, so this wouldn’t
even make sense.∗ What an observer actually calls ‘energy’ is the quantity conjugate
to the observer’s proper time. That is, the observer will consider a mode ∼ e−iωobs τobs
to have energy ωobs , in joules or similar energy units. The proper time of a uniformly
accelerating observer with acceleration a (and therefore Rindler position Robs = 1/a)
is
1
dτobs = Robs dη = dη ,
a
(5.18)
so the observer will see a mode e−iωη to have energy
ωobs = aω .
(5.19)
Accordingly, the temperature shown on an accelerating thermometer will be
Tobs =
a
.
2π
(5.20)
Aside: Transient acceleration
Strictly speaking, our discussion of accelerating observers assumes that the observer
has always been accelerating, and will continue accelerating forever. Only observers
who will continue accelerating forever actually have a Rindler horizon.
However, a temporarily accelerating observer will also see Unruh radiation. It does
not quite make sense to talk about a ‘temperature’ in this case because the observer’s
thermometer will not reach exact equilibrium in any finite time. When the observer
starts accelerating, there will be some transient effects, and then the observer will feel
thermal radiation; the thermometer will start to heat up, asymptotically approaching
temperature a/2π; and when the observer stops accelerating the thermometer will again
experience some transient effects, then radiate and cool back down to zero temperature.
So, as long as the acceleration lasts a long time compared to the equilibration timescale
tequil ∼ 1/T ∼ R0 , the Unruh temperature is still meaningful in this situation. On the
other hand, for short bursts of acceleration, our analysis does not apply. Instead we
would need to solve a time-dependent problem. This can be done using Feynman
∗
ie ds2 = dR2 − R2 dη 2 , so all the dimensions are carried by R; η is like an angular coordinate.
58
diagrams that describe emission/absorption of particles from an arbitrary worldline.
(There are many wrong papers on this topic. A correct, clear, and short paper that
also has a nice derivation of the Unruh effect from Green’s functions is: “Transient
phenomena in the Unruh effect,” Bauerle and Koning.)
5.3
Importance of entanglement
What does physics look like in the Rindler vacuum, |0iR ? To an accelerating observer,
it would look ordinary: this observer would detect no particles. A geodesic observer,
however, would detect observers, since this observer must notice that fields are not in
the Minkowski vacuum. As long as the geodesic observer is in the Rindler wedge, this
just looks like some particular excited state. However, timelike geodesics cannot stay
in the Rindler wedge forever — eventually they go through the Rindler horizon. The
Rindler vacuum state is singular at the horizon. That is, the energy density measured
by a geodesic observer diverges at the Rindler horizon. There is no ’beyond’ the horizon
in this state.
This makes sense. In the Rindler vacuum, there are no correlation between fields in
the left and right Rindler wedges:∗
R h0|φ(x1 )φ(x2 )|0iR
= 0 for x1 ∈ Rlef t ,
x2 ∈ Rright .
(5.21)
If there are no correlations, who’s to say that these wedges are actually ‘next to each
other’ ? In a sense they are not. Thus in the vacuum state, the Rindler wedge does not
extend beyond the horizon.
The key to obtaining a finite energy density on the Rindler horizon is to have a lot
of entanglement between the left and right Rindler wedges. In the exercise below you
will show explicitly how, in the Minkowski vacuum, the left and right Rindler wedges
are maximally entangled, much like the two spins in Bell’s thought experiment.† Any
state with smooth horizon must be highly entangled across the horizon.
∗
†
In this equation |0iR means the product vacuum where each Rindler wedge is in its vacuum.
http://en.wikipedia.org/wiki/Bell’s theorem
59
Exercise: Entanglement warm-up
Difficulty level: easy
Consider a quantum system consisting of two particles A and B, each with two states
|0i and |1i (which you can think of as spin-up and spin-down). Suppose the full system
is in the maximally entangled pure state
|ψi = |0iA |1iB + |1iA |0iB .
(5.22)
(This is sometimes called a Bell pair). Find the reduced density matrix ρA for particle
A. You will find a mixed state. Compute the entanglement entropy of this mixed state,
defined as
SA = − trA ρA log ρA .
(5.23)
Exercise: Thermofield Double
Difficulty level: conceptual
Consider a quantum system with Hilbert space H. Any mixed state ρ can be thought
of as a pure state in an enlarged system. That is, we can always add an auxiliary
Hilbert space H̃ and find a pure state
|Ψi ∈ H̃ ⊗ H
(5.24)
ρ = trH̃ |ΨihΨ| .
(5.25)
such that
This is called purifying the mixed state. In this problem you will show that Minkowski
space is a purification of Rindler space.
The Minkowski Hilbert space∗ factorizes† into two copies of the Rindler Hilbert space,
HM = H̃R ⊗ HR ,
(5.26)
which are the Hilbert spaces associated to the left Rindler x < 0 and right Rindler
∗
By ‘Minkowski Hilbert space’ we really mean Hilbert space of the theory on an infinite plane,
since Hilbert spaces are defined by the space on which a theory lives, not the spacetime. Similarly by
‘Rindler Hilbert space’ we mean the Hilbert space of the theory quantized on a half-plane.
†
This is not quite true due to UV divergences, but this doesn’t matter for this problem
60
x > 0.∗ In terms of the field data, this just means that a field in Minkowski space at
t = 0, φM (x), can instead be written as the pair (φ̃R , φR ) where φ̃R is a field on the
left Rindler half-plane, and φR is a field on the right Rindler half-plane.
(a) The Minkowski groundstate |0i is formally a functional that turns field data at
t = 0 into complex numbers. That is, the ground state wavefunction is
Ψ0 [φ̃R , φR ] = M hφ̃R , φR |0iM .
(5.27)
The subscript M means ‘Minkowski’, ie a state on the full space. Write down the path
integral that computes this wavefunction, and draw the corresponding picture along
the lines of the path integral pictures above.
(b) Now re-slice this same path integral using the Rindler Hamiltonian HRindler , which
generates Euclidean rotations ∂φ . That is, write an operator expression of the form
Ψ0 [φ̃R , φR ] = R hφ̃R | · · · |φR iR
(5.28)
and fill in the dots with an expression involving HRindler .
(c) We want to show that the Minkowski state is the same as the doubled Rindler state
|T F DiR⊗R ≡
X
e−βEn /2 |niR |ni∗R
(5.29)
n
where this is a sum over Rindler energy eigenstates, En is the Rindler energy, β = 2π is
the Rindler temperature, and ∗ means CPT conjugate. This is called the thermofield
double state.
To demonstrate this, check that the matrix elements of the state defined in (5.29) agree
with the ones you wrote above,
M hφ̃R , φR |T F DiR⊗R
= Ψ0 [φ̃R , φR ] .
(5.30)
To do you this you will need to note that the mapping from Minkowski states to Rindler
∗
Clearly HR = H̃R but the tilde will be useful to keep track of things.
61
⊗ Rindler states is
|φ̃R , φR iM → |φ̃i∗R |φR iR ,
(5.31)
where the conjugation is needed because time runs ‘backward’ in the left Rindler wedge.
What you have just shown is that the full Minkowski vacuum can be reinterpreted as
the thermofield double in two copies of Rindler space.
(d) Finally, check that tracing over the left Rindler Hilbert B space produces a thermal
state in the right Rindler Hilbert space A,
ρA ≡ trB |T F DihT F D| = e−2πHRindler =
X
e−2πEn |niR R hn|
(5.32)
n
This is an alternative derivation of the Unruh thermal state.
Reference: This problem is based on Maldacena’s thermofield double interpretation of
black holes in AdS/CFT [hep-th/0106112], which we will hopefully discuss later in the
course.
Comment: Another way to approach this problem is to think of the infinite-temperature
P
state n |niR |ni∗R as produced by a path integral over an infinitessimal strip in the
Euclidean plain, positioned along the negative τ -axis. Then the thermofield double
state is produced by evolving this by β/4 to the left and β/4 to the right, producing a
state on the τ = 0 line.
Exercise: State on an interval in 2d CFT
Difficulty level: a couple pages
In this exercise you will work out the state of a 2d conformal field theory in the vacuum,
when restricted to a finite interval. This is similar to the Unruh state, but on a finite
region instead of a semi-infinite plane (which in 2d would be a half-line).
At t = 0, we define region A to be the interval x ∈ (0, `), and region B is everything
else. Let z be a complex coordinate on 2d Euclidean space. The conformal mapping
z=−
w
w−`
62
(5.33)
maps the half-line in the z-coordinate to the interval x ∈ (0, `), where w = x + itE . In
the z coordinate, evolution of the half-line is generated by the rotational vector field ζ.
(a) Write ζ in the z coordinate,∗ then do the coordinate change (5.33) to write it in
the x, tE coordinates.
(b) The modular Hamiltonian, which generates the time evolution of the interval, is
then
Z
HA =
dx Ttµ ζ µ |tE =0 ,
(5.34)
A
where Tµν is the usual stress tensor. Write the integrand explicitly in terms of Ttt , x,
and `.
It follows that the quantum state on the interval is
ρA = e−2πHA .
(5.35)
(c) Sketch the vector field ζ on the Euclidean (x, tE ) plane.
(d) Continue tE → it, and sketch the vector field ζ in 2d Minkowski space (x, t).
5.4
Hartle-Hawking state
We have seen there is no unique vacuum state in quantum field theory. The same is
true on a black hole background. A natural state to consider, which is analogous to
the vacuum state we defined in Minkowski space, is a state prepared by a path integral
on the analytically continued Euclidean spacetime,
ds2 = (1 − 2M/r)dτ 2 +
dr2
+ r2 dΩ22 .
1 − 2M/r
(5.36)
with the imaginary-time identification τ ∼ τ + β. This spacetime only has r > 0, there
is no interior. The t = 0 slice of the Lorentzian spacetime is the τ = 0 slice of the
Euclidean spacetime, see figure 2.
∗
i.e., define z = z1 + iz2 and write ζ in terms of the two real coordinates z1 and z2 .
63
Figure 2: Schwarzschild spacetime. The Euclidean path integral produces a pure,
highly entangled state on the two-sided Lorentzian spacetime. The quantum state on
the right half of the Penrose diagram, where we live, is therefore mixed. This reduced
state is the Hartle-Hawking thermal state.
Sending τ → τ + β/2 takes us to the other side of the Penrose diagram in the maximal
analytic extension of Schwarzschild. This can be shown in detail using Kruskal coordinates. A simpler way to see this is to go to Rindler coordinates near the horizon.
By changing these Rindler coordinates to Minkowski-like coordinates good near the
horizon, we can continue through the horizon to the other side of the Penrose diagram.
So, just like in Rindler space, we get to the other side of the horizon by going half way
around the Euclidean circle.
This path integral prepares an entangled state on M̃ × M , the product of the left and
right Minkowski spaces. Just as in Rindler space, the reduced density matrix on our
spacetime M will be a mixed state,
ρHH = e−βH
(5.37)
where H is the ordinary Minkowski Hamiltonian associated to time translations ∂t .
This is called the ‘Hartle-Hawking state.’ It describes a black hole in equilibrium with
a bath of radiation outside the black hole.
This is not the only state we could consider. See note 3 for a discussion of other
possibilities. Hawking showed that a black hole formed by collapse will end up in the
‘Unruh state’, which is a state where the black hole radiates into a cold outside region.
Greybody factors
The Hartle-Hawking vacuum (5.37) is time-independent. This means that, in each
64
mode, the flux of outgoing Hawking radiation is equal to the flux of ingoing radiation.
A mode φk outside the black hole does not necessarily fall in; it is absorbed with
probability given by the absorption cross-section σabs (k). Therefore, the only way a
black hole can be in thermal equilibrium with a bath at temperature T is if the Hawking
emission measured at infinite is actually
hnk i =
1
σabs (k) .
eβw − 1
(5.38)
The extra factor is the ‘greybody factor’. We will probably calculate some greybody
factors later.
Aside: Cosmology
If the early universe is described by inflation, then it is the story of a slowly evolving
de Sitter spacetime. De Sitter spacetime is the Lorentzian continuation of a sphere.
That is, the metric of Euclidean de Sitter is just
ds2 = dΩ2D .
(5.39)
The equator of the S D is the t = 0 slice of global de Sitter space:
The state of quantum fields during inflation is responsible for present-day observables
including the primordial temperature fluctuations in the CMB, observed by experiments like COBE, WMAP, and Planck. Since there is no unique vacuum, we must
pick a state of the quantum fields in de Sitter. For various reasons,∗ we usually assume
this state is the so-called ‘Euclidean vacuum’, also called the ‘Bunch-Davies vacuum’
∗
Here are some reasons: (1) This state respects the symmetries of de Sitter; (2) At short distances,
this vacuum is the one in which comoving observers see no particles (ie it coincides locally with the
Minkowski vacuum); (3) at late times, due to the cosmological expansion, any state will dilute into
this state.
65
or various other things. This state is prepared by a Euclidean path integral on the
hemisphere, cut along the equator. Therefore this quantum state, unlike the HartleHawking state, has quite possibly already been observed experimentally.
Exercise: Decay of Schwarzschild
Difficulty: a couple pages
(a) A black hole in asymptotically flat spacetime loses energy via Hawking radiation.
If the initial mass is M , how long before the black hole radiates away completely?
(b) How heavy, in solar masses, would a black hole need to be for its lifetime to be the
age of the universe t ∼ 13 billion years?
(If such black holes exist, we might be able to observe the final moments of decay,
when a large burst of energy is released in Hawking radiation. Unfortunately there
is no particularly good reason to think they should exist, since black holes formed by
stellar collapse must have Minitial & a fewMsun .)
(c) What is the typical energy (in eV) of a particle emitted from a solar mass black
hole via Hawking radiation?
Exercise: Superradiance
Difficulty: a few lines
Rotating (Kerr) black holes are labeled by mass M and angular momentum J, or
equivalently by a temperature T and angular potential Ω. The spacetime is rotationally
invariant and stationary, so modes of a scalar field can be written φ ∼ e−iωt+imφ Sn (r, θ),
where n labels the solutions of given ω, m.∗ The Hawking decay rate of a rotating black
hole is
Γω,m,n =
1
eβ(ω−mΩ)
−1
σabs (ω, m, n)
(5.40)
(a) Take the zero-temperature limit of (5.40). (Hint: ω > 0 and m is any integer. The
∗
Actually, the wave equation fully separates, so in fact S(r, θ) = R(r)F (θ). This is surprising and
nontrivial, since the background has only two Killing vectors. Similarly, the geodesic equation on Kerr
has an ‘extra’ conserved quantity.
66
answer should not be trivial.)
(b) For this decay rate to make any sense, what can you conclude about σabs ?
Your conclusion is a phenomenon called ‘superradiance.’ It is a wave analogue of the
Penrose process discussed previously. In this exercise we took the Hawking formula
as our starting point, but the result is entirely classical – you would reach the same
conclusion by solving the wave equation on the Kerr background, and treating the black
hole scattering experiment as a 1d quantum mechanics barrier transmission problem.
Superradiance very efficiently converts rest mass to radiation energy. It is believed to
be responsible for the absurdly high luminosity of quasars: a single quasar consisting of
a highly rotating black hole surrounded by infalling matter has roughly the luminosity
of the entire Milky Way (1011 stars!).∗
∗
More accurately, a close cousin of superradiance involving magnetic fields. The details of how this
works are still largely unknown.
67
6
The Gravitational Path Integral
6.1
Interpretation of the classical action
In ordinary QFT, to do a path integral we first fix the spacetime manifold M , then
integrate over fields defined on M . We did the same thing in our discussion of Hawking
radiation. In quantum gravity, however, we must integrate over the geometry itself.
We are only allowed to specify the boundary conditions on the geometry as r → ∞,
just like for other fields. The gravitational path integral (in Euclidean signature) is
Z
Z=
−SE [g,φ]
DgDφ e
,
1
SE [g] = −
16πGN
Z
√
g (R + · · · ) + boundary terms ,
(6.1)
where φ denotes all the matter fields.
The meaning of this path integral depends on the boundary conditions, as usual. In
analogy to the QFT case, we define the thermal partition function Z(β) as the path
integral on a Euclidean manifold with the boundary condition that Euclidean time is
a circle of proper size β,
tE ∼ tE + β ,
gtt → 1,
at infinity .
(6.2)
Of course we cannot actually do the path integral. In fact, we don’t even really know
how to define it.∗ The best we can do is to approximate it by expanding around a
classical saddlepoint, i.e., a solution of the classical equations of motion:
Z(β) ≈ exp −SE [ḡ, φ̄] + S (1) + · · ·
.
(6.3)
The leading term, in which ḡ, φ̄ is a solution of the classical equations of motion, is the
semiclassical approximation to the path integral. This solution must of course obey
the correct boundary condition. The next term is the 1-loop term and is O(G0N ), and
the dots indicate higher-loop contributions.
We already know a solution with the correct boundary conditions: the Euclidean
∗
The situation in gravity is even worse than in ordinary QFTs, since the Euclidean action is not
bounded below.
68
Schwarzschild black hole. This is a classical saddlepoint with a Euclidean time circle of size β. Therefore, to leading approximation, the thermal free energy is the
Euclidean on-shell action:
log Z(β) ≈ −SE [ḡ] ,
(6.4)
with ḡ the Schwarzschild metric. (We have dropped φ̄ because no matter fields are
non-zero in the Schwarzschild background.)
This partition function can be used in all of the same ways as an ordinary thermodynamic partition function. For example, recall that log Z = S − βE, so the entropy and
energy are
S = (1 − β∂β ) log Z(β),
E = −∂β log Z .
(6.5)
We will see that these agree with the area law and the black hole mass.
A similar discussion applies with an angular potential and electric potential, but we
will stick to the Schwarzschild black hole to keep things simple.
Exercise: RN free energy
Difficulty: 2 lines
Using our previously calculated results for S and T (from the 1st law), and assuming
energy E = M , find the free energy of the Reissner-Nordstrom black hole.
Exercise: RN specific heat
Difficulty: 2 lines
Compute the specific heat of Reissner-Nordstrom.
6.2
Evaluating the Euclidean action
We will now do this explicitly, in Einstein gravity (i.e., no higher curvature corrections)
with zero cosmological constant. It is not as simple as computing R (which vanishes
69
for Schwarzschild!) and integrating over spacetime, since there are boundary terms to
worry about and infinities to regulate.∗ Although this is entirely classical, the procedure
to regulate divergences involves counterterms much like those in QFT; in fact we will
see later there is a direct link between these two apparently different divergences.
Gibbons-Hawking-York boundary term
The Euclidean action is computed by first cutting off the spacetime at some large but
fixed r = r0 . In the presence of a boundary we must add to the bulk Einstein action a
boundary term, called the Gibbons-Hawking-York term (once again setting GN = 1),
1
SE [g] = −
16π
√
Z
M
1
gR −
8π
Z
√
hK .
(6.6)
∂M
Here hij is the induced metric on the boundary ∂M , and the extrinsic curvature of
∂M is
Kij ≡ 12 Ln hij = ∇(i nj) ,
K = hij Kij ,
(6.7)
with n is the inward-pointing unit normal to ∂M .†
The Gibbons-Hawking-York term is needed for the action to be stationary around
classical solutions. The variation of the Einstein term has the schematic form
Z
δ
M
√
gR ∼
Z
Z
(eom)δg +
M
[A(g, ∂g)δg + B(g, ∂g)∂δg] ,
(6.9)
∂M
where ‘eom’ essentially means the Einstein tensor‡ and the boundary terms come from
integrating by parts. On a classical solution, the bulk term vanishes. If we impose
boundary conditions that fix the metric at r = r0 , then δg|∂M = 0, so the first boundary
term vanishes, but the boundary term involving ∂δg does not. The Gibbons-HawkingYork term fixes this problem. It is chosen so that the variation of the full action (6.6)
∗
This subsection follows Hawking’s chapter in General Relativity, an Einstein Centenary Survey,
Hawking and Ellis eds.
†
In the simple case that the boundary is, say, at fixed r, the induced metric hij = gij where i runs
over the transverse directions. That is all we will need. But more generally, the projector onto ∂M is
hµν = gµν − nµ nν
(as you can see by noting nµ hµν = 0), and then you must define intrinsic coordinate xi on ∂M .
√
‡
(eom)δg ∝ gGµν δgµν
70
(6.8)
has the form
Z
1
2
(eom)δg +
δSE [g] =
M
√
Z
hT µν δgµν .
(6.10)
∂M
We will return to this ‘stress tensor’ later, but for now the important thing is just that
the boundary term has been chosen to eliminate ∂δg. Thus δSE [ḡ] = 0 for variations
satisfying the boundary condition and ḡ satisfying the equations of motion.
Euclidean Schwarzschild Black Hole
The Euclidean Schwarzschild solution is obtained from the ordinary Schwarzschild
metric by sending t → −iτ ,
2M
dr2
ds = 1 −
+ r2 dΩ22 .
dτ 2 +
2M
r
1− r
2
(6.11)
What was the horizon r = 2M in Lorentzian signature is now just the origin of a polar
coordinate system, with angular coordinate τ identified as required for regularity at
the origin,
τ ∼ τ + 8πM .
(6.12)
Euclidean black holes are completely smooth solutions; they do not have an interior or
a singularity.
Now we want to evaluate the action. The bulk term vanishes, since the vacuum Einstein
equations set R = 0. The boundary term, evaluated on the surface r = r0 , is
√
hK = β(8πr0 − 12πM ) .
Z
(6.13)
∂M
This is infinite as we take r0 → ∞. The procedure to regulate this divergence∗ is to
add a ‘counterterm’ to the action,
1
SE [g] = −
16π
Z
M
√
1
gR −
8π
Z
∂M
√
1
hK +
8π
Z
√
hK0 ,
(6.14)
∂M
where K0 is the extrinsic curvature of the same boundary manifold ∂M , embedded in
∗
Caveat: this version of the procedure does not always work in asymptotically flat spacetime. As
far as I know there is no entirely satisfactory and unique prescription with zero cosmological constant.
Things are understood better in de Sitter space, which has finite volume, or in asymptotically anti-de
Sitter space, where a similar procedure always works and plays an important role in AdS/CFT.
71
flat spacetime. This is very similar to what we do in quantum field theory, but this
calculation is entirely classical. (We will see later that in anti-de Sitter space, there is
a direct connection between the two ideas). Note that the counterterm depends only
on data intrinsic to the boundary surface – it is not allowed to depend on ∂n h.
To compute the counterterm, we embed the boundary metric
ds2bdry = (1 − 2M/r0 )dτ 2 + r02 dΩ22
(6.15)
in flat space, by repeating the calculation for the flat geometry
ds2subtraction = (1 − 2M/r0 )dτ 2 + dr2 + r2 dΩ22 .
This gives∗
Z
√
hK0 = β(8πr0 − 8πM + O(1/r0 )) .
(6.16)
(6.17)
∂M
This eliminates the divergence (and changes the finite term!), giving our final answer
SE =
βM
= 4πM 2
2
(6.18)
Thus the thermal partition function, or leading approximation to the path integral, is
Z(β) = exp −4πM
2
β2
.
= exp −
16π
(6.19)
From this we can rederive the entropy and energy using standard thermodynamics,
S = (1 − β∂β ) log Z = 4πM 2
(6.20)
E = −∂β log Z = M
The entropy agrees with the area law S = Area/4.
Entropy and conical defects
We have just checked this for a special case, the Schwarzschild black hole, but this
∗
In this case you can get the same answer by just evaluating K with M = 0. However this does
not always work. The correct procedure is to subtract the curvature of a boundary surface of identical
intrinsic geometry, embedded in flat spacetime.
72
always works and agrees with the area law. Roughly, the reason it is proportional to
area is that we can think of the equation (1 − β∂β ) log Z as calculating the change
in the classical action produced by changing the imaginary-time identification. If you
smoothly deform a solution, then δSE = 0 by the equations of motion; but if you
R
introduce a defect, this contributes δSE = def ect (something). Going through the
details, you can derive Area/4.∗ This is also the easiest way to derive Wald’s formula,
which includes the corrections to the entropy from higher curvature terms in the action.
Exercise: Schwarzschild action
Difficulty: a page or two
Derive equations (6.13) and (6.18) using (6.7).
Exercise: Euclidean methods for the BTZ black hole
Difficulty: difficult, I suspect
Evaluate the on-shell action of the Euclidean BTZ black hole obtained by Wick-rotating
the metric (2.23). Check that you reproduce the correct entropy and energy. (Caveat!
What I called M in the metric is not the energy. The energy is E = M 2 /8.)
This calculation is similar to what we just did for asymptotically-flat Schwarzschild
black holes. Note that the bulk term no longer vanishes, R − 2Λ 6= 0. The full action,
including the counterterm, is
1
SE [g] = −
16π
Z
M
√
1
g(R − 2Λ) −
8π
Z
∂M
√
a
hK +
8π
Z
√
h.
(6.21)
∂M
Choose a to cancel the divergence; the remaining finite expression is the correct SE .
Reference: [Balasubramanian and Kraus, hep-th/9902121].
Comment: The counterterm depends on the dimensionality of spacetime. The simple
counterterm in (6.21) only works in AdS3 . In higher-dimensional AdS, there are more
R
available counterterms, for example ∂M R[h] (the intrinsic boundary curvature), and
these are also required to cancel all divergences.
∗
The clearest reference I know for this is section 3.1 of Lewkowycz and Maldacena, 1304.4926.]
73
74
7
Thermodynamics of de Sitter space
Inflation is the idea that the very early universe, at t . 10−32 s, is approximately de
Sitter spacetime. We will now take a detour to apply our methods to de Sitter space,
since in this situation they produce observable effects via the imprint of primordial
density perturbations on the CMB.
de Sitter is the maximally symmetric solution of the Einstein equations with a positive
cosmological constant. The metric is∗
ds2 = −dT 2 + `2 cosh2 (T /`)dΩ2D−1
(global coordinates) .
(7.1)
These coordinates, which cover all of de Sitter space, describe a sphere S D−1 that is
very large at T → −∞, contracts to a minimum radius ` at T = 0, then expands as
T → ∞. The length scale is set by the cosmological constant, Λ ∼ 1/`2 .
To draw the Penrose diagram, we need to make the range of T finite. This can be done
by defining
tan(η/2) = tanh(T /2`) .
(7.2)
In these coordinates, which run over η ∈ (− π2 , π2 ), the metric is
ds2 =
`2
−dη 2 + dΩ2D−1 .
2
cos η
(7.3)
For the Penrose diagram (which captures the causal structure but ignores distances)
we can ignore the overall factor, so this gives the diagram in figure 3. Note that de
Sitter space has an initial and final conformal boundary. (Although the diagram also
appears to have left and right boundaries, these are not really boundaries – at each
value of η space is a sphere, so those lines are just the north and south poles of the
sphere S D−1 .)
Vacuum
As usual, there is no unique vacuum. However we can follow our usual prescription,
∗
I will call these global coordinates, but in the context of cosmology this is usually called the ‘closed
slicing.’
75
Figure 3: Penrose diagram of de Sitter space.
as we did for Minkowski and Schwarzschild, and define a vacuum which is prepared by
a path integral on the Euclidean continuation of de Sitter. Euclidean de Sitter space
is just a sphere: send T → i`θ in (7.1), and you see that θ ∈ [−π/2, π/2] is the polar
coordinate on a sphere S D of radius `. It is shifted by π/2 from the usual definition,
so the surface T = θ = 0 is the equator of the S D . This equator is the minimal-size
spatial section of de Sitter space, an S D−1 of radius `. This is the T = 0 surface at the
middle of the Penrose diagram.
The Euclidean vacuum ∗ is the state prepared by a path integral on the hemisphere.
This path integral prepares a quantum state on the equator, which can then be evolved
in Lorentzian time. Here is a cartoon for this path integral:
(7.4)
Roughly speaking, ‘most’ states will eventually end up close to the Euclidean vacuum,
since the de Sitter expansion dilutes any excitations.
This vacuum state has in a sense been observed experimentally, since it leaves an
imprint in the CMB. I say ‘in a sense’ because it is not a very fine-grained or direct
test of the details of de Sitter space, but it is nonetheless the only good explanation
∗
Also called the ‘Bunch-Davies vacuum’ and various other things.
76
we have for those fluctuations.
7.1
Vacuum correlators
Now we will calculate the Gaussian fluctuations in de Sitter that are seen in the CMB,
using path integral methods.∗ We consider a free massless scalar field, in the ‘conformal
coordinates’ (7.3) in D = 4. (It turns out that scalar fluctuations of the metric reduce
to this problem.) The action is
Z
I∼
√
−g∇µ φ∇µ φ .
(7.5)
M
We want to compute the wavefunction for this scalar field, in the Euclidean vacuum.
This wavefunction is defined to be the transition amplitude
Ψ[φ0 ; t0 ] = hφ0 |e−iHt0 |0i .
(7.6)
This is a path integral over field configurations on the mixed spacetime (7.4). The
past boundary condition is regularity on the Euclidean sphere; the future boundary
condition is φ(Ω, t0 ) = φ0 (Ω).
In the WKB approximation the wavefunction is simply the on-shell action of a classical
solution satisfying these boundary conditions,
Ψ[φ0 ] ∼ eiI[φ] .
(7.7)
In fact, since this is a free field, this expression is exact. The on-shell action is a
pure boundary term, since we can integrate by parts in (7.5) then use the equations of
motion. Thus
Z
Ion−shell (t0 ) ∼
√
hφ∂n φ
(7.8)
Σ(t0 )
where Σ(t0 ) is the spatial slice at time t0 , ∂n is the derivative normal to this slice, and
h is the spatial metric.
At late times, the S D−1 spatial sphere in (7.3) is very large. When we look back at the
∗
This is based on Maldacena astro-ph/0210603.
77
CMB, we are looking at a tiny patch of this sphere. Therefore at late times we can
think of the spatial sphere as being essentially flat. That is, we can only see modes with
wavelength much smaller than `. This simplifies the calculation of the wavefunction
because at late times we can approximate the metric by
ds2 =
`2
(−dη 2 + d~x2 ) .
η2
(7.9)
This is enough to compute the wavefunction, since (7.8) is a pure boundary term at
t = t0 (η = η0 ).
~
For a plane wave φ0 = hk (η)eik~x , the wave equation at late times (i.e., using the metric
(7.9)) is
φ = 0
⇒
2
∂η2 hk (η) − ∂η hk (η) + ~k 2 hk (η) = 0 .
η
(7.10)
In order to find the wavefunction, we need to find the classical solution satisfying the
boundary conditions stated above. This is
φ = φ0k
(1 − ikη)eikη
.
(1 − ikη0 )eikη0
(7.11)
This satisfies all of our criteria: it solves the wave equation, and it is equal to φ0k at
time η = η0 . The last criterion was regularity on the Euclidean sphere – this is what
picks the solution going as e+ikη , rather than the other solution ∼ e−ikη . To see this,
note the pole of the Euclidean sphere is t = i`π/2, which corresponds to
η = 2 tan−1 (iπ/4) = i∞ .
(7.12)
Only the e+ikη solution is regular in this limit, so we’ve picked the correct solution,
corresponding to the Euclidean vacuum state. (This is very similar to the path-integral
calculation of the groundstate for the harmonic oscillator.) Although φ is a real field,
(7.11) is complex. This is fine: even though the path integral is over real field configurations, the stationary phase approximation can pick out a complex saddlepoint. It is
just a trick to compute the path integral.
78
Plugging (7.11) into (7.8),
Z
iI = i
d3 k `2 0
φ ∂η φ0k |η=η0 ∼
(2π)3 2η02 −k
d3 k `2 k 2
3
i − k + · · · φ0−k φ0k
(2π)3 2 η0
Z
(7.13)
This gives the wavefunction (exactly in a free theory) via (7.7). Knowing the wavefuncR
tion, we can calculate correlators using hφ2 i ∼ Dφφ2 |Ψ(φ)|2 . In detail, accounting
for the normalization of the wavefunction:
R
hφk φ−k i =
R 3
d p 2 3
Dφ φk φ−k exp − (2π)
`
p
φ
φ
p −p
3
R 3
R
d p 2 3
`
p
φ
φ
Dφ exp − (2π)
p −p
3
1 −3
k ,
2`2
R
2 R
2
where we used the Gaussian integral dzz 2 e−az / dze−az =
= (2π)3
(7.14)
1
2a
(and we’ve dropped
an overall momentum-conserving delta function).
The power law in (7.14) is what is meant by the statement that ‘inflation predicts a
scale-invariant spectrum of primordial scalar perturbations.’
7.2
The Static Patch
A single inertial observer travels on a geodesic which we might as well call the north
pole of S D−1 . Thus the worldline of an observer is the solid line on the right side of the
Penrose diagram. It is clear from the diagram that this observer is in causal contact
with only a subregion of de Sitter. This is because the universe is expanding very
rapidly, so you cannot communicate with someone beyond a certain critical distance:
the cosmological horizon. This is labeled ‘horizon’ in figure 3. Like Rindler space, the
position of the horizon depends on the observer.
The static patch is the region of de Sitter in causal contact with an observer sitting
at the north pole. This is the analogue of the Rindler patch. The coordinates on the
static patch are
ds2 = −(1 − r2 /`2 )dt2 +
dr2
dr2 + r2 dΩ2D−2 .
1 − r2 /`2
79
(7.15)
This looks like a black hole except, the static patch where an observer lives is the
inside, r < `, with a cosmological horizon at r = `.
Temperature
In the Euclidean vacuum, an observer in the static patch will see a temperature.∗ This
is the same reason we discussed above for Rindler space and then for the Schwarzschild
spacetime in the Hartle-Hawking state. The Euclidean path integral prepares a state
entangled between the left and right static patches, and when we trace over the hidden
region, the resulting density matrix is thermal. To find the temperature we can apply
the imaginary time periodicity trick. Starting from the static patch coordinates, define
r = `(1 − 2 ) and expand in to find:
ds2 ≈ 2`2 (d2 −
2 2
dt ) + · · ·
`2
(7.16)
This looks like Rindler, so the Euclidean continuation is regular only if t ∼ t + 2πi`.
Therefore the de Sitter temperature is
TdS =
1
.
2π`
(7.17)
This is the temperature that you will read on your thermometer if you are an inertial
observer in de Sitter. (Due to dark energy we are now at the start of a new de Sitter
epoch. The present-day de Sitter temperature is the Hubble scale, T ∼ 10−33 eV .)
Entropy
The area of the cosmological horizon is
Area = `2 V ol(SD−2 )
(7.18)
SD=4 = π`2 .
(7.19)
So in D = 4, S =area/4 gives
∗
See Gibbons and Hawking, “Cosmological event horizons, thermodynamics, and particle creation,”
1977.
80
7.3
Action
The Euclidean action of dSD is the action of S D . This is finite and there are no
boundary terms to worry about on a sphere, so it’s straightforward to calculate. In
D = 4, the Euclidean action is∗
1
IE = −
16π
`4
√
g(R − 2Λ) = −
16π
SD
Z
12
6
− 2
2
`
`
V ol(S4 ) = −π`2 .
(7.20)
Curiously, this is minus the entropy, i.e., the entropy is
S = log Z .
(7.21)
Here’s why: Recall the thermodynamic identity log Z = −βF = −βE + S. The energy
in GR is a pure boundary term (see next lecture!), so a compact space has E = 0. Thus
thermodynamics predicts log Z = S, and that’s exactly what we found in de Sitter.†
de Sitter is mysterious
We will not say much more about de Sitter space in this course. A big reason for this
is that we don’t have any UV-complete theory of gravity in de Sitter, like we do in
anti-de Sitter. We also have no clear answer to the question ‘What is the de Sitter
entropy?’ (Does it count the microstates of something?) Since we live in de Sitter, this
seems like a very important question.
Exercise: de Sitter in general dimensions
Difficulty level: straightforward, if using Mathematica to compute curvatures
∗
Given the proliferation of things called ‘S’ I will start using I for action
Caveat: In this calculation T = 1/(2π`), with ` a fixed parameter, so the temperature is not
actually a tunable variable. We can’t really make sense of thermodynamics if the temperature is not
a free variable. To really make sense of the discussion in this section, we need to add some additional
matter to de Sitter, which allows us to tune the temperature. Then there is a true 1st law, with the
temperature a tunable variable. This done in for example the nice de Sitter review hep-th/0110007.
You can think of what we’ve described here as the limit of that calculation where the extra energy is
set to zero.
†
81
Compute the entropy and the on-shell action of D-dimensional de Sitter space, and
verify the relation S = log Z.
82
8
Symmetries and the Hamiltonian
Throughout the discussion of black hole thermodynamics, we have always assumed
energy = M . Now we will introduce the Hamiltonian formulation of GR and show
how to define conserved charges associated to spacetime symmetries. The energy is a
special case, associated to time-translation symmetry. There are quicker ways to reach
the conclusion energy = M (see Carroll’s book), but we will take the more careful route
because it’s useful later.
8.1
Parameterized Systems
[References: The original paper is very nice and still worth reading, especially sections
1-3: “The Dynamics of General Relativity” by Arnowitt, Deser, Misner (ADM), 1962
(but available on arXiv at gr-qc/0405109). See also appendix E of Wald’s textbook,
and for full detail see Poisson’s Relativist’s Toolkit chapter 4.]
Time plays a special role in the canonical formulation of quantum mechanics, and in
the Hamiltonian approach to classical mechanics, since it is the independent variable.
In GR, time t is just an arbitrary parameter, and the dynamics are reparameterizationinvariant under t → t0 (t), since this is just a special case of diffeomorphisms. To see
how this fits into Hamiltonian mechanics we first consider a simple analog in quantum
mechanics.
Suppose we have a system with a single degree of freedom q(t) with conjugate momentum p, and action
Z
I=
dtL.
(8.1)
The Hamiltonian is the Legendre transform
H(p, q) = pq̇ − L(q, q̇)|p=∂L/∂ q̇ .
(8.2)
Hamilton’s equations of motion are
dp
∂H
=−
,
dt
∂q
dq
∂H
=
.
dt
∂p
83
(8.3)
The independent variable t is special. It labels the dynamics but does not participate
as a degree of freedom. In GR, time is just an arbitrary label – it is not special, and
the theory is invariant under time reparameterizations. To mimic this in our simple
system with 1 dof, we will introduce a fake time-reparameterizations symmetry. To
do this we label the dynamics by an arbitrary parameter τ , and introduce a physical
‘clock’ variable T , treated as a dynamical degree of freedom. So instead consider the
system of variables and conjugate momenta
q(τ ),
p(τ ),
T (τ ),
Π(τ )
(8.4)
where Π is the momentum conjugate to T . This is equivalent to the original 1 dof if
we use the ‘parameterized’ action
0
I =
Z
dτ (pq 0 + ΠT 0 − N R) ,
R ≡ Π + H(p, q) ,
(8.5)
where prime = d/dτ . Here N (τ ) is a Lagrange multiplier, which enforces the ‘constraint
equation’
Π + H(p, q) = 0 .
(8.6)
The action (8.5) is reparameterization invariant under τ̄ = τ̄ (τ ), since after all τ is just
a label that we invented. The Hamiltonian of the enlarged system is simply
H 0 = N (Π + H(p, q)) ,
(8.7)
so it vanishes on-shell due to the constraint equation!
To recap: we introduced time-covariance by adding a fake degree of freedom, and imposing a constraint. The resulting Hamiltonian vanishes on-shell, because it generates
τ -translations, which is just part of the reparameterization symmetry.
To reverse the procedure, i.e., to go from the parameterized action back to the ordinary
action with 1 dof, we plug in the constraint
0
I =
Z
dτ [pq 0 − H(p, q)T 0 ]
84
(8.8)
and then rewrite the dynamics in terms of the clock variable:
0
Z
I =
dT [pq̇ − H(p, q)]
(8.9)
where dot = d/dT . So we see that T is just the original physical time t.
The equation of motion for T is
T0 = N
∂
(Π + H)
∂T
(8.10)
But both T 0 and N are unspecified by the dynamics. For example we are free to pick
the ‘gauge condition’ T = τ , which corresponds to some particular choice of N (τ ).
8.2
The ADM Hamiltonian
GR is already a ‘parameterized system:’ the t coordinate is like our τ coordinate above,
and we will see that the Hamiltonian is very much like (8.7).
The canonical variables are
hij (~x, t),
πij (~x, t)
(8.11)
where hij are the space components of the metric, and πij are their canonical conjugates.
The full spacetime metric is parameterized as
ds2 = −N 2 dt2 + hij (dxi + N i dt)(dxj + N j dt) .
N =
(8.12)
p
−1/g tt is called the ‘lapse’ and N i = N 2 g ti is the ‘shift’. These are Lagrange
multipliers, just like N in our discussion above. They are not fixed by the dynamics,
but a choice of parameterization. In other words, any geometry can be sliced into
‘time’ and ‘space’ in a such a way that N and N i can be set to any functions you like.
They are called the lapse and shift because they correspond to our choice of how our
coordinates on a time-slice of fixed t = t0 are related to the coordinates on a time-slice
of fixed t = t0 + δt. The flow vector, which tells you the arrow of time from one slice
85
to the next, is∗
ζ µ = N uµ + N µ .
(8.13)
In the coodinates (8.12), ζ = ∂t , but we will treat N and N a as arbitrary parameters.
The action of GR, discussed above but now in Lorentzian signature, is
1
I=
16π
Z
√
1
d x −gR −
8π
M
4
Z
d3 x
√
−γ(K − K0 )
(8.14)
∂M
where K0 is the subtraction term (the extrinsic curvature of the boundary embedded
in flat spacetime). Recall that the on-shell variation is
δIon−shell =
1
2
√
d3 x −γT ij δgij ,
Z
(8.15)
∂M
where the ‘boundary stress tensor’ (aka Brown-York stress tensor) is
T ij =
1
K ij − γ ij K − background subtraction .
8π
(8.16)
After quite a bit of work† , the full off-shell action (8.14) can be written
Z
I=
h
i Z
ij
i
d x π ḣij − N H − N Hi −
4
M
d3 x
√
σuµ Tµν ζ µ ,
(8.17)
∂M
From here we can read off the Hamiltonian‡
Z
3
i
Z
d x (N H + N Hi ) +
H[ζ] =
Σ
d2 x
√
σuµ Tµν ζ ν ,
(8.18)
∂Σ
which is an integral over a spatial slice Σ.
Now to explain all these terms: H and Hi are called the Hamiltonian and momentum
constraints, which are essentially the G00 and G0i components of the Einstein equations
(see Wald for explicit formulae). These components of the equations of motion involve
only 1st time derivatives. They are called ‘constraints’ because if we think of GR as
an initial value problem – specify initial data, then evolve in time according to the
∗
Here N µ = hµa N a , where hµν = gµν + uµ uν , i.e., hµa is the projector onto a spatial slice.
See Brown and York, “Quasilocal energy and conserved charges derived from the gravitational
action, ” 1993, and also Poisson’s Relativist’s Toolkit, Chapter 4.
‡
The bulk term is called the ‘ADM Hamiltonian’. As far as I know, the boundary terms were first
derived by Brown and York, and by Hawking and Horowitz.
†
86
dynamical equations – these are constraints on the allowed initial data hij , ḣij at t = 0.
This is in contrast to other, dynamical equations of motion, which tell you how that
data evolves in time.∗ Finally uµ is the timelike unit normal on the boundary, with
u2 = −1.
A few remarks about our final answer (8.18):
• The bulk term vanishes on-shell due to the constraint equations. The boundary
term does not vanish in general. This is related to the fact that diffeomorphisms
acting on the boundary are ‘real’ dynamics, whereas diffeomorphisms away from
the boundary are just redundancies.
• We have written the Hamiltonian as a functional of the lapse and shift, since
the dynamics leaves ζ unspecified. This corresponds to a choice of time evolution. That is, the Dirac bracket† of the Hamiltonian with any function X of the
canonical variables is
{H[ζ], X} = Lζ X .
(8.19)
If we choose, for example, ζ µ = (1, 0), then this Hamiltonian generates time
evolution in the t-direction.
• The on-shell Hamiltonian looks just like the Hamiltonian of a 3-dimensional theory living on the boundary with a 3-dimensional stress tensor Tµν . We will see
that at least in AdS this is actually literally the case.
Exercise: Constraints in electrodynamics
Difficulty level: medium
Derive the Hamiltonian of electrodynamics. Start from the action I = − 41
∗
R
2
d4 xFµν
,
In electrodynamics, the action involves only first derivatives of At , so this is a Lagrange multiplier
like the lapse in GR. The Hamiltonian of electrodynamics has a term At C where C = ∇ · E − ρmatter
is the Gauss constraint.
†
The Dirac bracket is the Poisson bracket, but accounting for gauge symmetries which modify the
bracket acting on physical fields. The Dirac bracket is what becomes a commutator in the quantum
theory.
87
identify the canonical coordinates and conjugate momenta, and rewrite it like we did
for gravity in (8.17). Identify the Lagrange multiplier(s) and constraint(s).
In gravity, we gave a physical interpretation of the lapse and shift Lagrange multipliers
as a choice of foliation of spacetime. What is the analogous interpretation of At in
electrodynamics? (It might be useful to couple to a matter field to answer this.)
Reference:
Appendix E of Wald. But write your answers in terms of the vector
~ and B.
~
potential, not E
8.3
Energy
As usual, the numerical value of the Hamiltonian, evaluated on a solution, is the
energy. In GR we must specify a lapse and shift to define the Hamiltonian. The energy
is associated to time translations, so we identify the energy as the Hamiltonian with
N → 1 and N a → 0 at the boundary. For this choice, the surface deformation vector
µ
= (1, 0, 0, 0), so
ζ(t) has components ζ(t)
Z
E ≡ H[ζ(t) ]|on−shell =
√
d2 x σui Tit .
(8.20)
∂Σ
This is the usual (covariantized) expression for the energy in terms of the stress tensor.∗
We must impose boundary conditions to ensure that energy is conserved. It can be
shown that the Grµ components of the Einstein equations† are
αj
∇i T ij = −nα Tmatter
,
(8.21)
where n is the spacelike unit normal to the boundary. Therefore, if we impose the
boundary condition that matter fields go to zero fast enough as r → ∞, then the
∗
This equation agrees with other definitions of energy you may have seen, like the Komar formula,
whenever those definitions apply.
†
i.e., the ‘constraints’ in a radial slicing of the spacetime, which contain only first order rderivatives.
88
boundary stress tensor is conserved,
∇i T ij = 0 .
(8.22)
If in addition
∇(i ζj) = 0
r→∞
as
(8.23)
then the energy current j i = T ij ζ j is conserved, ∇i j i = 0. In this case the energy is
independent of what slice Σ we choose to evaluate (8.20):
0
Z
E(Σ) − E(Σ ) =
2
√
Z
i
2
d x σu Tit −
√
i
d x σu Tit =
Z
√
d3 x −γ∇i T ij ζj = 0 .
∂Σ0
∂Σ
(8.24)
The equation (8.23) is the Killing equation, so the conclusion is that energy is conserved
as long as (i) matter fields fall off fast enough near infinity, and (ii) ζ = ∂t is an
asymptotic Killing vector.
What about matter?
The expression (8.20) includes the contribution from matter. The constraints ensure
that the metric at infinity knows about any matter localized in the interior: the matter
backreacts on the metric, and therefore contributes at infinity. This is just like the
Gauss law in E&M.
8.4
Other conserved charges
Other asymptotic Killing vectors will similarly lead to conserved quantities. For example, if ζ = ∂φ satisfies (8.23), then we can define the conserved charge
Z
J=
√
d2 x σui Tiφ .
(8.25)
∂Σ
This is in fact the angular momentum, and agrees with all the usual formulae for
computing the angular momentum of a spacetime.
We could also define boost charges, and get the full Poincare group. This requires some
modifications, since in this discussion ζi was a vector within the fixed ∂M , whereas
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boosts act on ∂M . The results are similar.
8.5
Asymptotic Symmetry Group
We have seen that the bulk Hamiltonian vanishes, but there are boundary terms that
compute conserved charges. Now I will try to explain physically what is going on here.
Local diffs are fake. Global diffs are real.
GR is locally diff invariant, but it is not invariant under diffs that reach the boundary.
To see this from the action, just vary it under a general diff ζ. The Lie derivative for
a density is
√
√
δζ ( gf ) ≡ Lζ ( gf ) = ∇µ (f ζ µ ) .
(8.26)
Applying this the Lagrangian density of GR we see that it is only diff-invariant up to
a boundary term,
Z
√
δζ ( gL) =
Z
M
dAµ ζµ L .
(8.27)
∂M
This is important, so I’ll rephrase: General relativity is invariant under local diffeomorphisms. These are like gauge symmetries: fake symmetries, redundancies, that do
not change the physics and are just a convenient human invention to describe massless
particles. However it is not invariant under diffeomorphisms that reach the boundary.
The coordinates as r → ∞ are actually important and meaningful, like the coordinates
in a non-gravitational theory. A time reparameterization with compact support, i.e.,
t → t0 (t, x) such that t0 → t as r → ∞, is a local diff and involves no physics. A global
time shift t → t + 1 acts at infinity and is true time evolution.
The bulk terms in the Hamiltonian, i.e., the constraints, correspond to local diffs, and
the boundary terms correspond to diffs that reach the boundary. That is why the bulk
term vanishes on shell and the boundary term does not.
Certain diffs that reach infinity are actual symmetries. By ‘actual’ symmetries, I
mean symmetries that act on the space of states in the theory: they take one state
to a distinct but related state with similar properties, as opposed to gauge symmetries
which physically do nothing.
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Asymptotic symmetries in U (1) gauge theory
The precise version of all these statements is the formalism of asymptotic symmetries.
The definition of the asymptotic symmetry group is the group of symmetry transformations modded out by trivial symmetries,
ASG =
symmetries
.
trivial symmetries
(8.28)
The definition of a ‘trivial symmetry’ is one whose associated conserved charge vanishes.
Let’s consider electromagnetism as an example. The action I = − 14
R
d4 x(Fµν F µν +
µ
Aµ Jmatter
) is invariant under an infinite number of transformations,
δAµ = ∂µ Λ(x) ,
δφ = iΛ(x)φ ,
(8.29)
where the second term indicates the usual phase rotation on charged matter. These
are gauge symmetries. A local gauge symmetry, ie a transformation for which Λ(x) has
compact support, does not have any conserved charge associated to it. In fact despite
this infinite number of symmetries we know electromagnetism has only one conserved
quantity, the total charge
Z
3
Q∼
d
0
xJmatter
Z
∼
d2 x Ftr .
(8.30)
∂Σ
Σ
This is the conserved charge associated to the global U (1) rotation – it exists and is
conserved even in the un-gauged theory. Thus the global rotation is physical, while
local phase rotations are just redundancies.
The definition (8.28) of the asypmtotic symmetry group is the group of all transformations, mod gauge transformations with zero associated charge. Therefore in electromagnetism,
ASG = U (1)global .
(8.31)
Asymptotic symmetries in gravity
We will not go into depth on the ASG in gravity right now, but I will just mention
some facts. The ASG in gravity is generated by the conserved charges, which we
argued above are the charges associated to some special vector fields, including those
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for which ∇(i ζj) → 0 at infinity. In asymptotically flat spacetimes, ie spacetimes
approaching Minkowski space fast enough as r → ∞, these are simply the Killing
vectors of Minkowski space. Thus the asymptotic symmetry group of asymptotically
flat spacetimes is the Poincare group.∗
This notion is important, because general spacetimes have no isometries, and therefore no local conserved charges. (For example, there is no ‘energy’ conserved along
the geodesic of a probe particle.) Asymptotic symmetries allow us to define global
conserved quantities in this situation.
The Poincare algebra in 4D has 10 generators: 4 translations Pµ and 6 Lorentz generators Mµν . The generators obey the Poincare Lie algebra
[Pµ , Pν ] = 0
(8.32)
1
[Mµν , Pρ ] = ηµρ Pν − ηνρ Pµ
i
1
[Mµν , Mρσ ] = ηµρ Mνσ − ηµσ Mνρ − ηνρ Mµσ + ηνσ Mµρ .
i
(8.33)
(8.34)
If we just label the generators as V A for A = 1 . . . 10, then this is just a Lie algebra
C
i[V A , V B ] = f AB
CV
(8.35)
with some structure constants f AB
C . Each of these generators is associated to a Killing
vector of Minkowski space:
V A ↔ ζ (A)µ ,
A = 1 . . . 10 .
(8.36)
For example P µ ↔ ∂µ , Mtx ↔ t∂x + x∂t , etc. The Killing vectors obey the same
algebra, under the Lie bracket:
Cµ
.
[ζ A , ζ B ]µLB ≡ ζ Aν ∂ν ζ Bµ − ζ Bν ∂ν ζ Aµ = f AB
Cζ
(8.37)
(Here A is a label of which vector, and µ is a spacetime index.)
∗
This is true at spacelike infinity. The story at null infinity is much more subtle since, in non-static
spacetimes, gravitational radiation reaches null infinity and distorts the asymptotics. This leads to
what is called the BMS group, which is an active area of research.
92
Recall that conserved charges generate the action of the diffeomorphism under Dirac
brackets. That is, the charge
QA = H[ζ A ]
(8.38)
{QA , X}DB = Lζ A X .
(8.39)
generates
For this to be consistent with the algebra, the charges themselves must obey the same
algebra:
{QA , QB }DB = f ABC QC .
(8.40)
{H[ζ], H[χ]}DB = H [ζ, χ]LB + constant ,
(8.41)
In other words,
where we have allowed a constant ‘central charge’ term in the algebra of charges, since
this would still be consistent with the action of the generators on X (and actually does
appear in important examples).
Sometimes the ASG leads to surprises. A famous example is in anti-de Sitter space.
The isometry group of AdSD is SO(D − 1, 2). So a natural guess is that the asymptotic
symmetry group of asymptotically-AdS spacetimes is also SO(D − 1, 2). This is true
for D > 3 but wrong in D = 3, as shown by Brown and Henneaux. We will talk about
this more later.
8.6
Example: conserved charges of a rotating body
The linearized solution of GR that carries both energy and angular momentum is
2M
2M
4j sin2 θ
2
ds = − 1 −
dt + 1 +
(dr2 + r2 dΩ2 ) −
dtdφ .
r
r
r
2
(8.42)
This is, for example, the metric far away from a Kerr black hole, or a rotating planet.
We will compute the energy and angular momentum using the on-shell Hamiltonian
(8.18). Here it is again, after enforcing the constraints:
Z
H[ζ] =
√
d2 x σui Tij ζ j .
∂Σ
93
(8.43)
The energy is associated to ζ = ∂t and the angular momentum to ζ = ∂φ .
Kinematics
We want to compute T ij . This is a tensor living on ∂M , which is the surface r = r0 .
To define tensors on ∂M , we first compute the unit normal to the ∂M ,
r
nµ dxµ =
1+
2M
dr .
r
(8.44)
The full metric can be split into the normal and tangential parts as
gµν = γµν + nµ nν .
(8.45)
γνµ projects onto the boundary, since nµ γνµ = 0. The components γiµ for µ = t, r, θ, φ and
i = t, θ, φ can be used to turn spacetime tensors into boundary tensors, and vice-versa:
Vi ≡ γiµ Vµ .
(8.46)
The induced metric on ∂M is
2M
4j
2M
i
j
2
2
γij dx dx = − 1 −
dt +r0 1 +
(dθ2 +sin2 θdφ2 )− sin2 θdφdt . (8.47)
r0
r0
r0
The timelike unit normal to a constant-t hypersurface is
uµ dxµ = (−1 +
M
+ O(1/r2 ))dt .
r
(8.48)
(This could be used to define the induced metric from hµν = gµν + uµ uν and corresponding projector but we won’t need these to compute the charges.) Projecting the
timelike normal onto the boundary doesn’t change anything, we still have
ui dxi = (−1 +
M
+ O(1/r02 ))dt ,
r0
(8.49)
where remember i runs over the boundary directions xi = (t, θ, φ).
Finally, we need the volume element of the boundary at fixed time. The induced metric
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on ∂Σ is
σAB dxA dxB = r02 (1 +
with volume element
2M
)(dθ2 + sin2 θdφ2 ) ,
r0
√
2M
2
σ = r0 1 +
sin θ .
r0
(8.50)
(8.51)
Stress tensor
The extrinsic curvature of ∂M is
Kµν = −∇(µ nν) .
(8.52)
Kij = hµi hνj Kµν .
(8.53)
As a boundary tensor,
The trace of K is the same whether we use Kµν or Kij (check this!). It is
K=−
2
3M
− 2 + O(r0−3 ) .
r0
r0
(8.54)
Now we compute the stress tensor from its definition (ignoring the background subtraction for now), Tij = Kij − γij K. (I’ve rescaled it by 8π to unclutter notation, but
will put the 8π back in the Hamiltonian below.) It has components
Ttt = −
8M
2
+ 2 ,
r0
r0
Ttφ = −
5j sin2 θ
,
r02
Tθθ = r0 + M,
Tφφ = sin2 θ(r0 + M ) (8.55)
plus higher order terms O(M 2 /r02 ). (In this equation we are still ignoring the background subtraction, we will deal with that below.)
Energy
The energy is the on-shell Hamiltonian for ζ = ∂t . Putting it all together, we have so
far for the energy
Eunsub
1
=
8π
Z
2r0 sin θ = −r0 ,
∂M
where ‘unsub’ means we have not dealt with the background subtraction yet.
95
(8.56)
To do the background subtraction, we repeat the whole calculation on the flat spacetime
ds2sub
2M
=− 1−
r0
2M
dt + 1 +
(dr2 + r2 dΩ2 ) .
r0
2
(8.57)
This is a flat spacetime with the same intrinsic geometry on ∂M .∗
Going through all the steps again, the subtraction term is Esub = −r0 − M . Therefore
the final answer is
E=M ,
(8.58)
as expected.
Angular momentum
The angular momentum is the on-shell Hamiltonian for ζ = −∂φ .† There is no background subtraction necessary. We find
1
J=
3j
8π
Z
dθdφ sin3 θ = j .
(8.59)
∂Σ
∗
We could include the angular momentum term, but we can shift the time coordinate to make
it O(1/r02 ) and it does not contribute. Put differently, we really only need to embed ∂Σ into flat
spacetime, not all of ∂M , so this is only important for the energy and we can ignore the angular
momentum.
†
The minus sign here is the standard convention. It is related to the fact that a mode e−iωt+imφ
carries energy E = +ω and angular momentum J = +m.
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9
Symmetries of AdS3
This section consists entirely of exercises. If you are not doing the exercises, then read
through them anyway, since this material will be used later in the course. The main
goal of this section is derive the famous result of Brown and Henneaux on the central
charge of AdS3 . This was done in the 80s, using slightly different techniques from what
we’ll use here, and later came to play an important role in AdS/CFT, as we’ll see later.
9.1
Exercise: Metric of AdS3
Anti-de Sitter space is a constant-negative-curvature spacetime. It is the maximally
symmetric solution of Einstein’s equation with a negative cosmological constant. AdSD
can be realized as a hyperboloid embedded in a D + 1-dimensional geometry. In this
section we will talk about AdS3 , which is the hyperboloid
XA X A = −`2
(9.1)
where A = 0, 1, 2, 3 is an index in the space Minkowski2 × Minkowski2 , with metric
HAB dX A dX B = −dX02 + dX12 + dX22 − dX32 .
(9.2)
To find intrinsic coordinates on AdS3 , we just need to solve (9.1). One way to solve
this equation is by
X0 = ` cosh ρ cos t,
X1 = ` sinh ρ sin φ,
X2 = ` sinh ρ cos φ,
X3 = ` cosh ρ sin t .
(9.3)
(a) Check that this solves (9.1), and use (9.2) to find the induced metric on the hyperboloid.
Answer:
ds2 = `2 (− cosh2 ρ dt2 + dρ2 + sinh2 ρ dφ2 ) .
(9.4)
These are global coordinates on AdS3 . Although on the hyperboloid (9.1) we can see
from (9.3) that t is a periodic coordinate, when we say ‘AdS3 ’ we will always mean
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the space in which t is ‘unwrapped’, t ∈ (−∞, ∞) (the universal covering space of the
hyperboloid).
(b) Find the cosmological constant in terms of the AdS radius `.
9.2
Exercise: Isometries
AdS3 inherits the isometries of the embedding space that preserve the hyperboloid.
(For the same reason that the isometries of S 2 are inherited from rotations in R3 .) The
group of rotations+boosts in a 4d geometry with signature (+, +, −, −) is SO(2, 2), so
we expect this to be the isometry group of AdS3 . In this problem we’ll confirm this.
(a) As an example, consider the boost vector
V = X 1 ∂X 0 + X 0 ∂X 1
(9.5)
in the embedding space (9.2). This preserves the hyperboloid, since under X A →
X A − V A , the lhs side of (9.1) is unchanged to linear order (check this).
Write V as an isometry of AdS3 , in the coordinates (9.4). To do this, first define the
projection tensor
PµA =
∂X A
∂xµ
(9.6)
where xµ are the coordinates of AdS3 . This can be used to convert the 4-vector V A
into a tensor living on the hyperboloid,
χµ = PµA VA .
(9.7)
Find χµ , and check that it is a Killing vector of the metric (9.4).
This same procedure can be used to find all of the Killing vectors of AdS, but I will
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spare you the trouble. The answer, in a convenient basis, is
ζ−1 =
ζ0 =
ζ1
ζ̄−1
tanh(ρ)e−i(t+φ) ∂t + coth(ρ)e−i(t+φ) ∂φ + ie−i(t+φ) ∂ρ
(9.8)
1
(∂
2 t
+ ∂φ )
= 12 tanh(ρ)ei(t+φ) ∂t + coth(ρ)ei(t+φ) ∂φ − iei(t+φ) ∂ρ
= 12 tanh(ρ)e−i(t−φ) ∂t − coth(ρ)e−i(t−φ) ∂φ + ie−i(t−φ) ∂ρ
1
(∂
2 t
− ∂φ )
= 21 tanh(ρ)ei(t−φ) ∂t − coth(ρ)ei(t−φ) ∂φ − iei(t−φ) ∂ρ
ζ̄0 =
ζ̄1
1
2
Note that the subscripts here are just labels, not spacetime indices.
(b) Check that the vectors ζ−1 , ζ0 , ζ1 are Killing vectors.
(c) Now check that they obey the SL(2, R) algebra:
[L1 , L−1 ] = 2L0 ,
[L1 , L0 ] = L1 ,
[L−1 , L0 ] = −L−1 .
(9.9)
That is, the Killing vectors obey this algebra under Lie brackets, with an additional i,
for example
i{ζ1 , ζ−1 }LB = 2ζ0 ,
etc.
(9.10)
The barred zetas in (9.8) commute with the unbarred zetas, and form another SL(2, R)
algebra. Therefore the isometries of AdS3 form the algebra
SL(2, R)L × SL(2, R)R .
(9.11)
The subscripts mean ‘left’ and ‘right’, since the ζ’s involve only the ‘left-moving’ combination t + φ and the ζ̄’s involve the ‘right-moving’ combination t − φ.
Note that as a Lie algebra, of SO(2, 2) = SL(2, R) × SL(2, R). This is a special feature
of AdS3 . In general the AdSD isometry group is SO(D − 1, 2), which does not split
into two factors.
99
9.3
Exercise: Conserved charges
(a) Do the coordinate change
t± = t ± φ ,
ρ = log (2r) ,
(9.12)
and expand the metric (9.4) at large r. Show that to leading order
2
2
ds = `
dr2
− r2 dt+ dt−
r2
.
(9.13)
These are called Poincaré coordinates, and in fact this metric is an exact solution of
Einstein’s equation – it covers a subregion of AdS3 called the Poincaré patch.
A spacetime is called asymptotically AdS if it approaches (9.13) as r → ∞.∗
(b) Consider the asymptotically AdS spacetime
2
2
ds = `
dr2
− r2 dt+ dt−
2
r
+ h++ (dt+ )2 + h−− (dt− )2 + 2h+− dt+ dt− ,
(9.14)
where the h’s are arbitrary functions of t+ and t− but independent of r. We will
compute the boundary stress tensor (Brown-York tensor) and use it to define the
energy and other conserved charges in AdS3 .
The boundary stress tensor is defined as the variation of the on-shell action
T ij ≡ √
2 δSon−shell
−γ δγij
(9.15)
where the action is
1
S[g] =
16π
Z
M
√
1
−g(R − 2Λ) +
8π
√
a
−γK +
8π
∂M
Z
Z
√
−γ .
(9.16)
∂M
For the bulk term and Gibbons-Hawking term, we can use our formulae from flat space
given in previous lectures. The last term is a counterterm, which takes the place of the
∗
Specifically, the subleading components of the metric must have a certain fall-off at large r. These
conditions are basically chosen so that the Hamiltonian can be defined.
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‘background subtraction’ we did in flat space. This leads to
T ij =
1 ij
K − Kγ ij + ãγ ij .
8π
(9.17)
Choose the counterterm coefficient ã so that Tij is finite as the cutoff surface r0 → ∞.
Compute T++ , T−− , and T+− to first order in the perturbation hij in the limit r0 → ∞.
Reference: Balasubramanian and Kraus, hep-th/9902121.
(c) Compute the energy of the spacetime (9.14). It is defined as
1
E=
`
Z
2π
√
dφ σui Tij ζ j
(9.18)
0
where ui is the timelike normal to a fixed-t slice, and ζ = ∂t . (The overall 1/` is a
convention, necessary due to the fact we are using dimensionless coordinates.)
(d) An example of an asymptotically AdS spacetime is the BTZ black hole,
2
ds = `
2
dr2
+ r2 dφ2
−(r − 8M )dt + 2
r − 8M
2
2
.
(9.19)
Check that for this spacetime
E=M .
(9.20)
To use your results of the previous problem in this calculation you must first change
coordinates to put it in the form (9.14). In particular you will need to redefine r0 = r0 (r)
to eliminate the perturbation to grr .
(e) Compute the energy of global AdS, by keeping the subleading terms in the coordinate transformation (9.12) and plugging them into your formula for the stress tensor.
(Hint: the answer is negative. That’s OK, this is just a choice of zero for energy.)
Comment: We’ve focused on the energy, but we could compute conserved charges
corresponding to all the other Killing vectors in exactly the same way.
101
10
Interlude: Preview of the AdS/CFT correspondence
The rest of this course is, roughly speaking, on the AdS/CFT correspondence, also
known as ‘holography’ or ‘gauge/gravity duality’ or various permutations of these
words. AdS/CFT was conjectured by Maldacena in a famous paper in 1997. A full
understanding of Maldacena’s motivations and results, and the huge body of work to
follow, requires some string theory, but AdS/CFT itself is independent of string theory
and we will not follow this route. Instead we will ‘discover’ AdS/CFT by throwing
stuff at black holes. In fact, this parallels the historical discovery of AdS/CFT in
1996-1997, though we will obviously take a shorter path. Our starting point will be a
black-hole-like solution in 6 dimensions, which might seem umotivated, so the purpose
of this interlude is to describe where we are headed, so you know we are doing this for
a good reason.
10.1
AdS geometry
Anti-de Sitter space is the maximally symmetric solution of the Einstein equations
with negative cosmological constant. We worked out the metric of AdS3 in global and
Poincaré coordinates in the previous section. For general dimension AdSd+1 , the metric
in global coordinates is
ds2 = `2 − cosh2 ρdt2 + dρ2 + sinh2 ρdΩ2d−1 .
(10.1)
To find the Penrose diagram, we can extract a factor of cosh2 ρ and then define a new
coordinate by
dσ =
dρ
⇒ σ = 2 tan−1 tanh(ρ/2) .
cosh ρ
(10.2)
As ρ runs from 0 to ∞, ρ runs from 0 to π/2. Each value of t, σ is a sphere S d−1 . Therefore the Penrose diagram looks like a solid cylinder, where ρ is the radial coordinate of
the cylinder, and t, Ω are the coordinates on the surface of the cylinder.
Unlike flat space, the conformal boundary (usually just called ‘the boundary’) of AdS
is timelike. From the Penrose diagram, we can see that massless particles reach the
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boundary in finite time t. (Massive particles cannot reach the boundary; they feel an
e2ρ potential if they try to head to large ρ.)
The metric of the Poincaré patch is
`2
ds = 2 dz 2 − dt2 + d~x2 ,
z
2
(10.3)
where ~x = (x1 , . . . , xd−1 ). In these coordinates, the boundary is at z = 0. These
coordinates cover a wedge of the global cylinder. You can check this for AdS3 using
the coordinate transformations derived in the previous section.
10.2
Conformal field theory
A conformal field theory (CFT) is a QFT with a particular spacetime symmetry, conformal invariance. Conformal invariance is a symmetry under local scale transformations. We will discuss this in detail later. For now I will just mention that one
consequence of conformal symmetry is that correlation functions behave nicely under
coordinate rescalings x → λx. Correlation functions of primary operators (which are
lowest weight states of a conformal representation) obey
hO1 (x1 )O2 (x2 ) · · · On (xn )i = λ∆1 +∆2 +···+∆n hO1 (λx1 ) · · · On (λxn )i
(10.4)
where ∆i is called the scaling dimension of the operator Oi . This (together with
rotation and translation invariance) implies for 2pt functions
hO(x)O(y)i ∝
1
.
|x − y|2∆
(10.5)
The simplest example of a CFT is a free massless scalar field, where for example in
4d hφ(x)φ(y)i = (x − y)−2 . A massive free field is not conformal, since m shows up
in correlation functions and spoils the simple power behavior. This is generally true –
CFTs do not have any dimensionful parameters, so there can be no mass terms in the
Lagrangian. However the converse is not true, since there are theories with no mass
terms in the classical theory are not necessarily conformal. For example in massless
QCD, scale symmetry is broken in the the quantum theory so the theory acquires a
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dimensionful parameter via dimensional transmutation.
There are also very nontrivial interacting conformal field theories. We will discuss a
couple of examples later.
10.3
Statement of the AdS/CFT correspondence
The AdS/CFT correspondence is the an exact relationship between any∗ theory of
quantum gravity in asymptotically AdSd+1 spacetime and an ordinary CFTd , without
gravity. This relationship is called a duality. It is holographic since the gravitational
theory lives in (at least) one extra dimension. The theories are believed to be entirely
equivalent: any physical (gauge-invariant) quantity that can be computed in one theory
can also be computed in the dual. However, the mapping between the two theories can
be highly nontrivial. For example, easy calculations on one side often map to strongly
coupled, incalculable quantities on the other side.
It is often useful to think of the CFT as ‘living at the conformal boundary’ of AdS.
Indeed, the CFT lives in a spacetime parameterized by x = (t, ~x), whereas gravity
fields are functions of x and the radial coordinate ρ. And when we discuss correlation
functions of local operators we will see that a CFT point x corresponds to a point
on the conformal boundary of AdS. But it is not quite accurate to say that the CFT
lives on the boundary, for two reasons. First, we should not think about having both
theories at once; we either do CFT or we have an AdS spacetime, never both at the
same time. Second, the CFT is dual to the entire gravity theory, so in a sense it lives
everywhere.
The two theories are commonly referred to as ‘the bulk’ (i.e., the gravity theory) and
‘the boundary’ (ie the CFT).
In this course we will mostly restrict our attention to two types of observables in
AdS/CFT: thermodynamic quantities and correlation functions.
∗
Some people might obect to the word ‘any’ here. To be safe, we could say ‘any theory that we
know how to define in the UV and acts like ordinary gravity+QFT in the IR.’
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Thermodynamics
The mapping between thermodynamic quantities on the two sides of the duality is
simply that they should be equal, for example the thermal partition functions obey
Zcf t (β) = Zgravity (β) .
(10.6)
Here Zcf t = Tr e−βHcf t is the ordinary thermodynamic partition function of a QFT.
Zgravity is the quantity whose semiclassical limit we discussed above, related to the
on-shell action of a black hole,
Zgravity (β) = e−SE [g] + · · · .
(10.7)
For the exact relation (10.6) we must in principle include all the quantum corrections
to this semiclassical formula.
Correlation functions
The goal of the next couple lectures is to derive the dictionary that relates CFT
correlators to a gravity calculation. We will give the exact prescription later, but
here is the general idea. Each field φi (ρ, x) in the gravitational theory there is a
corresponding operator Oi (x) in the CFT.∗ The mass of φ determines the dimension
of O. CFT correlation functions can be computed on the gravity side by computing a
gravity correlator of φ, with the points inserted at the boundary:
hO1 (x1 ) · · · On (xn )icf t ↔ “ lim ”hφ1 (ρ, x1 ) · · · φn (ρ, xn )igravity
ρ→∞
(10.8)
The limit is in quotes because actually we need to rescale by some divergent factors
that we’ll come to later.
Top down, bottom up, and somewhere in between
AdS/CFT is general, we do not need to refer to a particular theory of a gravity or
a particular CFT. However it is often useful to have specific theories in mind, with
∗
Here x = (t, ~x) denotes all d dimensions of the CFT and ρ is the radial coordinate.
105
detailed microscopic definitions. For example: Type IIB string theory on AdS5 × S 5 is
dual to N = 4 supersymmetric Yang-Mills in 4d.
Super-Yang-Mills is a particular CFT with a known Lagrangian. Although IIB string
theory is not defined non-perturbatively (except via this duality), it has many known
microscopic ingredients. Calculations in these two specific theories can be compared
in great detail.
There are other microscopic examples — deformations of this one, and different versions
in different dimensions, with different types of dual CFTs. All of them (as far as I
know) come from brane constructions in string theory. This is often called the ‘top
down’ approach to AdS/CFT.
Another approach is to simply assume that we have a CFT with some low-dimension
primaries with a particular pattern, and perhaps with some assumptions about the
symmetries and conserved charges of the theory. This is more in the spirit of effective
field theory and is often called ‘bottom up.’ In many cases we can also include information about the UV completion of the CFT (i.e., very high dimension operators) in this
approach so it actually goes beyond effective field theory, but without every specifying
the actual Lagrangian of the CFT.
Both approaches are important. Often calculations that can be done in one approach
are impossible in the other, or calculations first done microscopically turn out to have
more general and possibly more intuitive explanations via effective field theory.
106
11
AdS from Near Horizon Limits
Anti-de Sitter space appears in the near horizon region of extremal black holes. In this
section we will describe how the near-horizon limit of extremal Reissner-Nordstrom is
AdS2 × S 2 . Since the case d = 1 (i.e., AdS2 ) is a special case of AdS/CFT that we
would like to mostly avoid, we then discuss the 6d black string. This solution has a
near-horizon AdS3 which will serve as our main example for AdS/CFT.
11.1
Near horizon limit of Reissner-Nordstrom
Here is the 4d Reissner-Nordstrom black hole again, from (2.10):
ds2 = −f (r)dt2 +
dr2
+ r2 dΩ22 ,
f (r)
f (r) = 1 −
2M
Q2
+ 2 .
r
r
(11.1)
Recall that this solution is restricted to M > Q (we assume Q > 0) by cosmic censorship. In general, the near horizon geometry is approximately Rindler ×S 2 , as discussed
above. But something special happens in the extremal limit,
M =Q.
(11.2)
In this limit the horizon is a double zero, f (r) = (1 − Q/r)2 . So the inner and outer
horizons coincide, r+ = r− = Q, and the Hawking temperature (2.19) is zero.
To take the near horizon limit of the extremal black hole, we define
r = Q(1 + λ/z),
t = QT /λ
(11.3)
where λ is an arbitrary parameter. Plugging this into the extremal metric and taking
the limit λ → 0 with z, T, θ, φ held fixed gives the spacetime
ds2 =
Q2
2
2
−dT
+
dz
+ Q2 dΩ22 .
2
z
(11.4)
The metric (11.4) is AdS2 × S 2 . This is the near horizon region of the original black
hole, since if λ → 0 with z held fixed, r → Q. Recall that this spacetime is supported
107
by some nontrivial electric field. Applying the same procedure to the field strength
gives a constant electric field in the near horizon region.∗
λ has disappeared entirely from the solution. This means that (11.4) (together with the
constant electric field) is actually by itself a solution to the Einstein-Maxwell equations.
This did not happen for the non-extremal case: there, the spacetime was only approximately Rindler near the horizon, and Rindler ×S 2 does not solve the Einstein-Maxwell
equations.
Geometry of the Near Horizon Region
The key difference between the extremal and non-extremal near horizon limits is that
the near horizon region of an extremal black hole is infinitely long. To see this let us
calculate the distance to the horizon along a fixed-t slice in the general metric (11.1),
from some arbitrary point r0 > r+ :
Z
r0
D=
dr
p
f (r) ∼ −M log(r+ − r− ) ∼ M log
r+
1
QTH
(11.5)
where TH is the Hawking temperature (2.19). This diverges as TH → 0. (The ∼ means
we are dropping constants and the contribution of the r0 limit to the integral, which
doesn’t matter.) The long region for small TH
Global coordinates
The near-horizon metric we found in (11.4) is AdS2 in Poincaré coordinates. This
covers only a patch of the full AdS spacetime. Similarly, we considered only one patch
of the Reissner-Nordstrom spacetime. The full global AdS2 includes many Poincaré
patches, and each patch gives the near-horizon region of a different patch of the global
Reissner-Nordstrom. This is illustrated in the Penrose diagrams in figure 4.
What about the sphere?
We are only drawing the Penrose diagrams for AdS2 , but the geometry is in fact
AdS2 × S 2 . Actually the sphere does not affect the conformal boundary. Since the
∗
In the original coordinates, Frt = Q/r2 . In the new coordinates, after scaling λ → 0, FzT =
−Q/z 2 . This does not look constant in these coordinates, but if we define σ = 1/z then FσT = Q is
a constant electric field.
108
Figure 4: Penrose diagrams for extremal Reissner-Nordstrom and AdS2 . The AdS2 Penrose diagram is a ‘zoomed in’ version of the RN diagram which includes only the near
horizon region. Unlike higher-dimensional AdS, AdS2 has two conformal boundaries,
which are the blue lines on the left and right. In the RN diagram, the coordinates are
degenerate near the horizon so these boundaries are drawn as dashed blue lines slightly
away from r = r+ . The black hole horizon in RN (red) is the same as the Poincaré
horizon in AdS2 . The Poincaré patch of AdS2 is shaded green in both diagrams.
109
prefactor in front of AdS2 blow up near the boundary, after a conformal rescaling the
sphere just drops out. So points on the conformal boundary are labeled only by T , not
by T, θ, φ.
Near horizon as a low-energy limit
A particle on a geodesic has a conserved energy-at-infinity
E ≡ −pt = f (r)
dt
.
dτ
(11.6)
For a particle in the near-horizon region, this is strictly zero as λ → 0. So from the
point of view of an observer at infinity, everything in the near-horizon region is infinitely
redshifted. Similarly, a wave in the near-horizon region ∝ e−iωnear T has zero frequency
as measured from infinity, since
ωnear T ∼ λωt ∼ 0 · t .
11.2
(11.7)
6d black string
Unfortunately AdS2 is the runt of the AdS/CFT litter. It is very interesting in its
own right but quite different from other dimensions (since CFT1 does not really make
sense) so not suitable for our purposes. We will focus on AdS3 instead, which appears
in the near horizon limit of a 6d black string (among other things). In the exercises
you will treat the other most popular example of AdS/CFT which involves AdS5 .
The 6d black string is similar to a black hole, but with horizon topology S 3 × S 1 . The
6d metric is∗
r02
2
2
2
ds = (f1 f5 )
−dt + dφ + 2 (cosh σdt + sinh σdφ)
r
2
dr
1/2
2
2
+ (f1 f5 )
+ r dΩ3 .
1 − r02 /r2
2
−1/2
∗
(11.8)
My favorite references on this solution and the discussion to follow are Kiritsis’s textbook, section
12.7, and MAGOO hep-th/9905111. Note that I will not distinguish between the Einstein-frame
metric and string-frame metric in this discussion; they differ by just a constant in the near-horizon
region which can be absorbed into the definition of `.
110
where
f1 = 1 +
r12
,
r2
f5 = 1 +
r52
,
r2
(11.9)
and φ ∼ φ + 2πR is compact. I will not write the other fields but there are some
nontrivial scalars and gauge fields that can be found in the references. This solution
carries two charges, called Q1 and Q5 , related to r1 and r5 . (In string theory these count
the number of D1 branes and D5 branes.) It also carries a momentum proportional to
r0 sinh σ along the φ direction, as you can guess from the boost term cosh σdt+sinh σdφ.
Finally r0 is the position of the horizon and measures the deviation from extremality.
−1
|r=r0 .
To see this, note the surface gravity is proportional to ∂r grr
This is called the D1-D5-P black string. (Often in the literature you will find that it
is dimensionally reduced to 5d along the φ direction, so it becomes a 3-charge black
hole.)
AdS3 in the near horizon of the extremal black string
Far away, i.e., r 0, this geometry is just R4 × S 1 . Now we will look at the decoupling
near horizon limit. This the same type of near horizon limit that we did for extremal
RN, so it produces an exact, infinite-volume solution of the equations of motion.
First we consider the case with zero momentum. The extremal D1-D5 with P = 0 is
obtained by setting r0 = 0,
ds2 = (f1 f5 )−1/2 (−dt2 + dφ2 ) + (f1 f5 )1/2 (dr2 + r2 dΩ23 ) .
(11.10)
The horizon is at r = 0. To take the near-horizon limit, we define
`2 = r1 r5
(11.11)
scale
r → λ`r ,
t → t`/λ ,
φ → φ`/λ ,
(11.12)
and send λ → 0. This has the effect of just dropping the 1 in fi = 1 + ri2 /r2 , so
ds2near
2
=`
dr2
2
2
2
+ r (−dt + dφ ) + `2 dΩ23 .
r2
111
(11.13)
This is the geometry AdS3 × S 3 , where the curvature radii of AdS and of the sphere
are equal.
Near-extremal D1-D5-P
Now let us take a different near-horizon limit of (11.8) where we simultaneously scale
r → 0 as we scale the black string towards extremality, r0 → 0. In this limit, r0 cosh σ
stays finite, so this is an extremal limit with finite momentum.
Starting from (11.8) we scale
r → λ`r , t → t`/λ , φ → φ`/λ , r0 → λ`r0 .
(11.14)
The resulting metric is
ds2near
2
=`
dr2
2
2
2
2
−r dt + 2
+ r dφ + r0 (cosh σdt + sinh σdφ) + `2 dΩ23 . (11.15)
r − r02
2
2
The term in brackets is in fact a 3d black hole, called the BTZ black hole. To see this
in more standard BTZ coordinates, define the parameters
w+ = r0 cosh σ,
w− = r0 sinh σ
(11.16)
and do the coordinate change
2
r2 = w2 − w−
.
(11.17)
The resulting metric is
dw2
ds2near
w + w − 2
2
2
= −h(w)dt +
+ w dφ +
dt + dΩ23 ,
2
2
`
h(w)
w
where
h(w) =
2
2
(w2 − w+
)(w2 − w−
)
.
w2
(11.18)
(11.19)
The w, t, φ part of this metric is a 3d black hole carrying mass and angular momentum,
with horizons at w± . Setting w− = 0 and w+ = 8M gives the J = 0 BTZ that was
used in some examples earlier in the course.
112
Exercise: AdS5 as near horizon limit
Difficulty: easy
Consider the 10D metric
ds2 = f −1/2 (−dt2 + d~x2 ) + f 1/2 (dr2 + r2 dΩ25 ) .
where
f =1+
r34
.
r4
(11.20)
(11.21)
r3 a constant parameter and ~x a coordinate on 4d space R4 . This metric is an extremal
black brane. (A black brane is like a black hole, but the horizon is a plane instead of a
sphere. In string theory, this solution is the geometry corresponding to a stack of Q3
D3 branes, where Q3 is a conserved charge of this solution, related to r3 .)
Show that the near-horizon geometry is AdS5 × S 5 .
Exercise: Near horizon Kerr
Difficulty: A little messier, use Mathematica!
The metric of the 4D Kerr black hole is
ds2 = −
ρ2
1
∆(r)
2
2
(dt−a
sin
θdφ)
+
dr2 +ρ2 dθ2 + 2 sin2 θ(adt−(r2 +a2 )dφ)2 , (11.22)
2
ρ
∆(r)
ρ
where
∆(r) = r2 + a2 − 2M r ,
ρ2 = r2 + a2 cos2 θ ,
(11.23)
and −M < a < M . This describes a rotating black hole with mass M and angular
momentum
J = aM .
(11.24)
Find the value of M (as a function of J) where this black hole is extremal. Then find
the near horizon geometry of the extremal Kerr.
Hint: To find a regular near-horizon metric, you must go to a rotating coordinate
system that corotates with the black hole horizon, ψ = φ − Ωt. After going to this
rotating coordinate system the calculation is similar to what we did for RN.
Reference: Bardeen and Horowitz, [hep-th/9905099]. (Also [0809.4266].)
113
12
Absorption Cross Sections of the D1-D5-P
We will now throw a scalar field at a near-extremal D1-D5-P black string. In the
process we will rediscover AdS/CFT. The metric is (11.8), where we now assume
r0 r1 , r5 .
(12.1)
This leads to a low Hawking temperature TH . (We also assume cosh σ, r1 /r5 ∼ O(1).)
Our goal is to calculate the absorption cross-section of a scalar field with low energy,
ωr5 1 .
(12.2)
We will assume the scalar χ has zero momentum around the φ direction and on the
3-sphere. It is convenient to define
TL =
1 r0 eσ
,
2π r1 r5
TR =
1 r0 e−σ
,
2π r1 r5
(12.3)
which will turn out to be left and right moving temperatures in the dual CFT. These
are related to the Hawking temperature by
2
1
1
=
+
.
TH
TL TR
12.1
(12.4)
Gravity calculation
The wave equation χ = 0 for a scalar field of the form χ = e−iωt R(r) in the metric
(11.8) is
where
f=
r2
1 + 12
r
h d 3d
2
hr
+ω f R=0
r3 dr
dr
r52
r02 sinh2 σ
1+ 2
1+
,
r
r2
(12.5)
h=1−
r02
.
r2
(12.6)
This is now basically a 1d quantum mechanics problem. To get some intuition for this
scattering process, define
1
χ(r) = p
ψ(r) .
r(r2 − r02 )
114
(12.7)
In these variables, the wave equation is
d2
− 2 + V (r) χ = 0
dr
(12.8)
where V (r) is easy to find and plot, but annoying to write down. It looks like a well
near the horizon r = r0 , falls off at infinity, and has a lump somewhere in between.
This looks just like an ordinary Schrodinger equation, so we are just scattering through
a potential.
To compute the absorption cross-section, we need to solve the wave equation and compare the coefficients of the incoming, transmitted, and reflected waves. The strategy
is to solve the equation approximately in the ‘near’ and ‘far’ regions, and match these
solutions together somewhere in the middle. The near and far regions are defined by
r r0
far:
r r1,5 ,
near:
(12.9)
r 1/ω
(12.10)
These regions overlap in the ‘matching region’ r0 rm r1,5 .
The general solution of the wave equation in the far region is a linear combination of
Bessel functions,
Rf ar = r
−3/2
r
πωr
[AJ1 (ωr) + BY1 (ωr)]
2
(12.11)
The general solution in the near region is
h
i
Rnear = Ãh−i(a+b)/2 + B̃h+i(a+b)/2 2 F1 (−ia, −ib, 1 − ia − ib, h)
(12.12)
with
a=
ω
,
4πTR
b=
ω
.
4πTL
(12.13)
The boundary condition is that the wave is purely ingoing at the horizon r = r0 . This
sets B̃ = 0. Then we expand both Rnear and Rf ar in the matching region:
Γ(1 − ia − ib)
+ O(r02 /r2 )
Γ(1 − ia)Γ(1 − ib)
√
π
≈ A √ ω 3/2 + B-terms
2 2
Rnear ≈ Ã
Rf ar
115
(12.14)
(12.15)
We have not written the B terms because they are messy, but we will use conservation
of flux to fix B later. Matching the terms in (12.14) gives
r
πω 3 A
Γ(1 − ia − ib)
= Ã
.
2 2
Γ(1 − ia)Γ(1 − ib)
(12.16)
The Wronskian of the 2nd order wave equation is interpreted as the conserved flux,
F≡
1 3 ∗
hr R ∂r R − cc ,
2i
dF
=0.
dr
(12.17)
We would like to compare the incoming flux at infinity to the transmitted flux entering
the horizon. The far solution, expanded near infinity, is
Rf ar ≈
1 iωr
−3πi/4
−iπ/4
−iωr
3iπ/4
iπ/4
e
Ae
−
Be
+
e
Ae
−
Be
.
2r3/2
(12.18)
Thus the incoming flux is
2
A
Fin = −ω .
2
(12.19)
Using the same formula to calculate the flux through the horizon gives the absorbed
flux
Fabs = −r02 (a + b)|Ã|2 .
(12.20)
The ratio of absorbed flux is (exercise!)
Rabs =
Fab
ω 4 π 2 (r12 r52 )
eω/TH − 1
.
=
Fin
4
(eω/2TL − 1)(eω/2TR − 1)
(12.21)
Greybody factors
This is the greybody factor that appears in Hawking emission, up to a factor. The
factor is required since the relation between spherical waves that we considered and
plane waves is
e−iωz = K
e−iωr
Y000 + · · ·
r3/2
where Y000 is the s-wave spherical harmonic on S 3 . The constant is K =
(12.22)
p
4π/ω 3 .
Therefore the absorption cross section for a plane wave is
σabs = |K|2 Rabs .
116
(12.23)
This is the greybody factor.
Exercise: Far and near Hawking temperatures
Difficulty: straightforward
(a) Calculate the Hawking temperature of (11.8) (with σ = 0, i.e., no rotation).
(b) Now calculate the Hawking temperature of the BTZ black hole that appears in the
near horizon, (11.18) (again with σ = 0, so w− = 0).
Note that when we took the near-horizon limit of the near-extremal string, we sent
r0 → 0. So any finite temperature of the BTZ is actually zero temperature as viewed
from asymptotically flat infinity. There is an infinite redshift between the near horizon
region and infinity.
Exercise: Scattering of a massive scalar
Difficulty: difficult
The wave equation for a massive scalar is
χ = m2 χ .
(12.24)
In this problem we will derive the absorption cross section of a low-energy massive
scalar on the near-extremal black string.
(a) Derive the full radial wave equation from (12.24), in the black string geometry
(11.8) (but with σ = 0).
(b)Find the ‘near-region’ wave equation by starting with your answer to part (a) and
assuming r r1,5 and rω 1.
(c) Show that your near-region wave equation is identical to the massive wave equation
on the BTZ ×S 3 geometry,
ds2near
2
=`
2
−(r̃ −
r02 )dt̃2
dr̃2
+ 2
+ r̃2 dφ̃2 + dΩ23
r̃ − r02
117
.
(12.25)
(Where the tilded coordinates are proportional to the original coordinates.)
(c) Find the ingoing solution of the wave equation in the near region. Do not bother
with the far-region solutions, since these are messy and nothing interesting happens in
the far region.
Hint: Mathematica cannot solve this wave equation without some coaxing. To simplify
it, first change variables so h = 1 − r02 /r2 is your independent variable. Then define
√
R(h) = (1 − h)a hb ψ(h), with a = 21 (1 + 1 + `2 m2 ) and b = −iω`2 /2r0 . Then Mathematica can solve it. This is also the best method to solve it by hand (i.e., first strip
out the singular points, then reduce the result to a standard hypergeometric equation).
(d) Show that in the matching region r0 r r1,5 , the field behaves as
Rnear ≈ Sr−d+∆ + F r−∆
(12.26)
where d = 2,∗ S and F are numbers (possibly functions of ω), and
2
m = ∆(d − ∆) ,
d
∆= +
2
r
d2
+ m2 `2 .
4
(12.27)
(e) If we think of the S term as the source, or ingoing term, and F as the response, or
outgoing term,† argue that the absorption cross section (of the near region) is proportional to the imaginary part of the ratio,
Pabs ≡ Im
F
S
(12.28)
and compute Pabs . (Including the far region too would just contribute some overall
uninteresting factors.)
The correct answer looks like
Pabs = k sinh(2πn)|Γ(
∗
∆
+ in)|4
2
(12.29)
I’ve only written d so that the result is true in higher-dimensional AdSd+1 /CFTd . In this problem
always set d = 2.
†
The words ‘ingoing’ and ‘outgoing’ are not quite accurate here since these are power-law solutions,
not traveling waves, in the matching region. But they have a similar interpretation.
118
where k = k(r0 , ∆) is a simple constant you should find, and n ≡ `2 ω/(2r0 ).
(f ) Define the retarded Green’s function
GR =
F
.
S
(12.30)
This measures the response of the field to adding a source. (The relation (12.28) is a
version of the optical theorem for this Green’s function.)
Find GR in the high-frequency limit ω/TH 1. (This is the correlator in the extremal
limit, where temperature goes to zero.) You should find a power law. This power-law
behavior at short distances is the hallmark of a conformal field theory.∗
(g) The zero-temperature 2pt function of a 2d CFT is
hO(x+ , x− )O(0)i = |x|−2∆ = (x+ x− )−∆
(12.31)
where ∆ is the scaling dimension of the operator. Take the 2d Fourier transform,
Z
G(ωL , ωR ) ∼
dx+ dx− eiωL x
+ +iω
Rx
−
(x+ x− )−∆ .
(12.32)
Don’t worry about the coefficient; we only care about the power law, so you can do this
Fourier transform by dimensional analysis. Check that for ωL = ωR = ω, your answer
agrees with part (f). Therefore, the quantity ∆ that we introduced in the process of
the solving the wave equation is equal to a CFT scaling dimension.
Exercise: Quasinormal modes
The scattering modes that we found above are modes that obey a single boundary
condition: ingoing at the horizon. Such modes have a continuous spectrum. A quasinormal mode is a mode that obeys two boundary conditions: ingoing at the horizon,
and outgoing far away from the black hole. These have a discrete spectrum. They are
quasi -normal instead of normal because they decay (as flux falls into the black hole) so
the the discrete frequencies have imaginary parts. If you perturb a black hole from the
vicinity of the horizon, the ‘ringdown’ is (roughly) described by quasinormal modes.
∗
Why did we have to take the high-energy limit to see this? The answer is that the temperature
introduces a scale; correlators in a CFT are only scale-invariant in the vacuum.
119
The quasinormal modes of BTZ are modes which are ingoing at the horizon and have
S = 0 in (12.26).
(a) Find the spectrum of of quasinormal modes ωn for a massless scalar in BTZ.
(b) If you did the previous exercise, then use your solution of the near-region wave
equation to find the spectrum of quasinormal modes ωn for a massive scalar in BTZ.
120
13
Absorption cross section from the dual CFT
Now we will reproduce the absorption cross section (12.21) using holography. This requires introducing some elements of conformal field theory. We will be more systematic
about CFT and about the AdS/CFT correspondence later, for now we are just going
to work this example in full detail as an illustration.
13.1
Brief Introduction to 2d CFT
(References: Polchinski’s String Theory Ch 2 is a brief introduction. For a more
detailed systematic introduction to 2d CFT, see chapters 4-6 (especially chapter 5) of
the (highly recommended!) book Conformal Field Theory by Di Francesco et al.
Consider a 2d QFT on the Euclidean plane R2 , with coordinates x1 and x2 . It is very
convenient to use the complex coordinates
z̄ = x1 − ix2 .
(13.1)
ds2 = (dx1 )2 + (dx2 )2 = dzdz̄ .
(13.2)
z = x1 + ix2 ,
We take the flat metric on the plane,
2d Conformal Transformations
A conformal transformation is a coordinate transformation that leaves the metric unchanged, up to an overall rescaling:
ds2 = dzdz̄ → eσ(w,w̄) dwdw̄ .
(13.3)
First we want to find what type of coordinate changes have this special property. To
this end, consider an arbitrary coordinate change z = f (w, w̄), z̄ = f¯(w, w̄) where f¯ is
the complex conjugate of f . The metric in (w, w̄) coordinates is
2
ds =
∂f
∂f
dw +
dw̄
∂w
∂ w̄
¯
∂f
∂ f¯
dw +
dw̄ .
∂w
∂ w̄
121
(13.4)
For this to have the form (13.3) we must impose
∂f ∂ f¯
∂f ∂ f¯
=
=0.
∂w ∂w
∂ w̄ ∂ w̄
(13.5)
This is equivalent to the condition that f is a holomorphic function,
f¯ = f¯(w̄) .
f = f (w),
(13.6)
Thus conformal transformations in two dimensions are equivalent to holomorphic coordinate changes. The conformal group is the group of holomorphic maps. This is infinite
dimensional, since you need an infinite number of parameters to specify a whole function. Note that this is not the case in higher dimensions; the conformal group in d > 2
dimensions is the finite-dimensional group SO(d, 2).
Mapping the plane to the cylinder
A very important conformal transformation is the mapping of the z-plane to the wcylinder. The mapping is
z = e−iw/R ,
z̄ = eiw̄/R .
(13.7)
The w coordinate labels a cylinder, since if we take w → w + 2πR we get back to where
we started. That is, w is identified,
w ∼ w + 2πR .
(13.8)
This circle is a circle of constant magnitude on the z plane. (Draw the pictures for
yourself.) If we split w into real coordinates,
w = σ1 + iσ2 ,
(13.9)
then σ1 ∼ σ1 + 2πR is the circle and σ2 is infinite. Negative values of σ2 correspond to
small circles on the z plane, and larger values of σ2 correspond to increasingly larger
circles on the z plane.
122
Classical CFT
At the classical level, a QFT has conformal symmetry if the action is invariant under
conformal transformations. For example consider the action of a free massless scalar
Z
S=
¯
d2 z∂φ∂φ
(13.10)
where we use the notation
∂¯ = ∂z̄ .
∂ = ∂z ,
(13.11)
Perform the infinitesimal coordinate change z → w(z) and it is easy to check that
the Jacobian in the measure cancels the factors that show up from ∂φ =
dw
∂ φ.
dz w
On
the other hand the free massive scalar is not conformally invariant. This illustrates a
general feature of conformal field theories: they do not have any dimensionful parameters. Dimensionful parameters set a scale and therefore are not compatible with the
scale transformation z → λz, which is part of the conformal group (in any number of
dimensions).
Quantum CFT
Classical conformal invariance does not necessarily imply quantum conformal invariance. This is familiar from QCD (setting all quark masses to zero) — this theory
is classically scale invariant, but to define the quantum theory we must introduce a
regulator, and this leads to the dimensionful QCD scale ΛQCD with important physical
consequence (like confinement), so QCD is certainly not scale-invariant or conformally
invariant at the quantum level. From now on when we say ‘CFT’ we mean at the
quantum level.
Primary operators
The local operators of a CFT must transform covariantly. Primary operators ∗ transform with a simple rescaling,
0
O (w, w̄) =
dw
dz
−h ∗
dw̄
dz̄
−h̄
O(z, z̄)
(13.12)
These are often called primary fields. The names are interchangeable. But remember that in
CFT, a ‘field’ is not necessarily a fundamental field that appears in the Lagrangian and is integrated
over in the functional integral. For example in the free massless scalar, ∂φ is a primary field.
123
where (h, h̄) are called the conformal weights. Another common notation is
s = h − h̄ ,
∆ = h + h̄,
(13.13)
where ∆ is the scaling dimension and s is the helicity. ∆ is the weight under a constant
rescaling (x1 , x2 ) → (λx1 , λx2 ), ie under
δz = λz,
δz̄ = λz̄
(13.14)
the operator transforms with a factor of λ−∆ . s is the helicity because it is the weight
under a rotation (x1 , x2 ) → (x1 − λx2 , x2 + λx1 ), ie under
δz̄ = −λz̄
δz = λz,
(13.15)
the operator transforms with a factor of λ−s . The absolute value |s| = |h − h̄| is the
spin of the operator. (This is just the usual definition of spin, so for example in free
field theory it corresponds to the number of Lorentz indices on a field.
Descendant operators are operators that you get from primaries by acting with conformal transformations. For example, ∂O(z, z̄) is a descendant of O(z, z̄). The transformation law for descendants is more complicated than (13.12) but is completely fixed
by symmetry.
All local operators in a CFT are either primary or descendant. This ensures that
correlation functions transform covariantly under the conformal group. For example,
the 2-point function on the plane must have the form
hO1 (z1 , z̄1 )O2 (z2 , z̄2 )i =
C12
(z1 − z2 )2h (z̄1 − z̄2 )2h̄
(13.16)
h̄ = h̄1 = h̄2 .
(13.17)
where
h = h1 = h2 ,
C12 is a constant, related to the normalization of the field. The two-point function
vanishes if the conformal weights of the two fields are different.
124
The path-integral definition of hO1 (z1 , z̄1 )O2 (z2 , z̄2 )i in (13.16) is (up to normalization)
Z
hO1 (z1 , z̄1 )O2 (z2 , z̄2 )i =
DΦO1 (z1 , z̄1 )O2 (z2 , z̄2 )e−S[Φ]
(13.18)
where Φ stands for the fundamental fields of the theory.∗ Recall from our discussion
of Euclidean path integrals that the path integral on a half-plane prepares the vacuum
state. Therefore in operator langauge,
hO1 (z1 , z̄1 )O2 (z2 , z̄2 )i = line h0|O1 (z1 , z̄1 )O2 (z2 , z̄2 )|0iline ,
(13.19)
where |0iline is the vacuum state of the theory on an infinite line (which you can think
of as the Im z = 0 axis).
13.2
2d CFT at finite temperature
Remember from our discussion of Euclidean path integrals that QFT at finite temperature in Lorentzian signature is related to Euclidean QFT on a cylinder, with periodic
imaginary time. Now we will see this relation very explicitly in CFT.
Mapping to the cylinder via w = iR log z, and applying the transformation law (13.12)
to (13.16), we can easily find the cylinder correlation function
hOcyl (w1 , w̄1 )Ocyl (w2 , w̄2 )i ∼
R−2h
sin
w1 −w2
2R
R−2h̄
2h
sin
w̄1 −w̄2 2h̄
2R
.
(13.20)
(The ‘cyl’ subscript is usually dropped, so functions of w are just assumed to be cylinder
operators.)
Exercise: Conformally invariant 2-point functions
(a) Prove (13.16).
∗
Often we do not have a Lagrangian for a CFT, and there is no useful notion of the ‘fundamental’
fields. However, path integral manipulations are still useful. Even in non-Lagrangian theories we
never get into trouble by pretending that there are some fundamental fields defining the functional
integral.
125
(b) Derive (13.20), including the missing coefficient.
Note two things about this correlator: First, it is invariant under the cylinder periodicity w1 ∼ w1 + 2πR.∗ Second, it has the same short-distance singularity as the plane
correlator (13.20), i.e.,
hOcyl (w1 , w̄1 )Ocyl (w2 , w̄2 )i =
C12
(w1 − w2 )2h (w̄1 − w̄2 )2h̄
as w1 → w2 .
(13.21)
This is always true in QFT: the short-distance behavior is fixed by vacuum correlation
functions. (In fact these two conditions fix the function (13.20) uniquely, assuming
some behavior at infinity, so we do not even strictly need the exponential mapping to
derive (13.20).)
From the Lorentzian point of view, the cylinder correlator (13.20) can be interpreted
different ways. To go to Lorentzian signature, write w = σ1 + iσ2 where σ1,2 are real
coordinates. If we think of σ2 as ‘time’, then the Wick rotation to Lorentzian signature
is σ2 = it. In this case the circle coordinate σ1 remains a circle in Lorentzian signature,
so this Wick rotation gives the Lorentzian theory on the Lorentzian cylinder S 1 × Time.
This Wick rotation has nothing to do with finite temperature.
To get the finite temperature theory, we instead Wick-rotate by setting σ1 = it. Thus
w → i(t + x),
w̄ = i(t − x) .
(13.22)
(Note that in Lorentzian signature, w and w̄ are no longer complex conjugates.) This
means that the the theory is periodic in imaginary time t ∼ t + 2πiR. Comparing to
the finite-temperature periodicity t ∼ t + iβ with β = T −1 , we see that our Euclidean
CFT is related to a finite-temperature CFT at temperature
β = T −1 = 2πR .
∗
(13.23)
There are some subtleties with branch cuts making this statement that we’ll ignore for now, and
it relies on the fact that (−1)2(h−h̄) = 1 since operators must have integer or half-integer spin.
126
From (13.20), this means the finite-temperature Lorentzian correlator in CFT is∗
Gβ (t − i, x) =
Tr e−βH O(t − i, x)O(0, 0)
∼ (−1)h+h̄
13.3
(πT )2h
(πT )2h̄
.
sinh(πT (t + x))2h sinh(πT (t − x))2h̄
(13.24)
Derivation of the absorption cross section
We now return to the derivation of the absorption cross-section (12.21). Recall that we
scattered a low-energy quantum from the near-extremal black string. The near horizon
region relevant to this calculation was a BTZ black hole in AdS3 (times S 3 ). We will
set TL = TR = TH for simplicity, which corresponds to setting the parameter σ = 0
in the black string metric. From the point of view of the near-horizon, this sets the
angular momentum of the BTZ black hole to zero.
In the gravity calculation (12.21), we found
σabs ∼ coth
ω
4TH
.
(13.25)
Now the claim is as follows:
We can replace the near-horizon geometry by 1+1d CFT at temperature T = TH , living
on a fictitious ‘membrane’ at the boundary of AdS3 .
This boundary was the matching location in our gravity calculation, ie some value of
r in the range r0 r r1,5 .
Which CFT is it?
We will only match the temperature dependence. The overall factor can also be
matched by this method, up to a constant. We will not need to specify which CFT
we are actually considering, we will just need some general properties of the CFT like
the value of the temperature, and the existence of an operator with certain conformal weights. The microscopic definition of the particular CFT depends the particular
∗
Setting a convenient normalization, and introducing an i to keep track of operator ordering.
Recall that the finite-temperature correlator is defined by ordering in Euclidean-time, so this sets the
order of the operators in the trace as written.
127
theory of quantum gravity. The only known microscopic CFTs are the ones coming
from string theory, since that is our only candidate theory of quantum gravity, but in
principle there could be other CFTs corresponding to other UV completions of gravity.
In the string theory examples, where the microscopic definition of the CFT is known,
it is also possible to match the coefficient in the absorption calculation and it comes
out correctly.
The interaction term
We want to scatter a scalar field against the CFT. We will assume that the bulk scalar
field χ couples to a CFT operator O, thus adding to the CFT an interaction term
Z
Sint =
dtdxO(t, x)χ(t, x, r = 0) .
(13.26)
In this expression O is a CFT operator and χ(r = 0) — the value of the bulk field
at the fictitious membrane where the CFT lives — is treated as a classical source.
We will assume that the space direction in the CFT is unwrapped, so we call it x ∈
(−∞, ∞) (previously called φ), though the S 1 version can also be done with some extra
assumptions about the CFT. We also assume the source couples weakly to the CFT so
that the interaction term (13.26) can be treated perturbatively.
Absorption rate
When we computed the absorption cross section, we assume
χ = e−iωt R(r) ,
so Sint ∝
R
(13.27)
dtdxO(t, x)e−iωt . The transition amplitude from an initial state |ii to a
final state |f i is given by Fermi’s Golden Rule, as the matrix element of the interaction
Hamiltonian
Z
Mi→f ∼ hf |
dtdxO(t, x)e−iωt |ii .
(13.28)
The total absorption rate at temperature β is computed by summing this over final
states, and averaging over initial states with a thermal ensemble,
Γabs ∼
X
e
−βEi
Z
dt1 dx1 dt2 dx2 e−iω(t1 −t2 ) hi|O(t2 , x2 )|f ihf |O(t1 , x1 )|ii
i,f
128
(13.29)
The sum over |f i is just the identity, so up to an overall factor of volume (which you
can think of as the momentum-conserving delta function δ(0)),
Z
Γabs ∼
dtdxe−iωt
X
e−βEi hi|O(t, x)O(0, 0)|ii .
(13.30)
i
This sum is the definition of the thermal 2-point function,
Z
Γabs ∼
dtdxGβ (t − i, x)e−iωt
(13.31)
This thermal correlator was calculated in (13.24). To take the Fourier transform, use
the integral
Z
−iωy
dye
(−1)
h
πT
sinh [πT (y ± i)]
2h
=
ω 2
(2πT )2h−1 ±ω/2T e
Γ(h + i
) . (13.32)
Γ(2h)
2πT
First take the Fourier transform assuming indepdendent left and right momenta
h+h̄
Gβ (ωL , ωR ) = (−1)
Z
dtdxe−iωL (t+x)−iωR (t−x)
(πT )2h
(πT )2h̄
sinh(πT (t + x))2h sinh(πT (t − x))2h̄
and then set ωL = ωR = ω. The absorption rate is given by the difference of absorption
and emission. These correspond to two different i prescriptions (Exercise: why?). So
finally
σabs ∼ Γabs − Γemit
Z
∼
dtdxe−iωt [G(t − i, φ) − G(t + i, φ)]
∼ 2
ω (2πT )2(h+h̄)−2
Γ(h + i ω )Γ(h̄ + i ω )2
sinh
2T
4πT
4πT
Γ(2h)Γ(2h̄)
(13.33)
(13.34)
(13.35)
This matches the gravity answer (13.25) if we set
h = h̄ = 1
(13.36)
and use the identity |Γ(1 + ix)|2 = πx/ sinh(πx). Why should the weight be (13.36)?
For now, we just pick them so the answer works out. In general the weights depend
on the mass and spin of the bulk field, and (13.36) is the correct choice for a massless
129
bulk field. We will treat this more systematically below.
13.4
Decoupling
The upshot of the last few sections is that





Far-region gravity
+
gravity in AdS3 × S 3










=
Far-region gravity
+



 .

CF T2 on AdS3 boundary
In the gravity calculation, we assumed near-extremal but not exactly extremal. This
retained some coupling between the near-horizon degrees of freedom, and the fields
in the asymptotically flat far region. Similarly, in CFT, we assume a weak coupling
between gravity fields and CFT fields.
If we take TH → 0, the far region and near regions decouple. This is Maldacena’s
decoupling limit. In this limit we can completely drop the asymptotically flat part of
the calculation, and we are left with the (3d version of the) AdS/CFT correspondence:
gravity in AdS3 × S 3
=
CF T2 on AdS3 boundary .
(13.37)
CFTs are UV-complete, so this duality defines not only low-energy effective gravity,
but a UV-complete theory of gravity on AdS3 × S 3 .
From now on, we will just forget about the ‘far region.’ It is not needed, and is rarely
used in modern AdS/CFT.
130
14
14.1
The Statement of AdS/CFT
The Dictionary
Choose coordinates
ds2 =
`2
(dz 2 + dx2 )
z2
(14.1)
on Euclidean AdSd+1 , where x is a coordinate on Rd . The boundary is at z = 0.
We showed above that scattering problems in gravity map to correlation functions
in CFT. In this relation the boundary value of the bulk field acted as a source for
a CFT operator. This is generalized by the following statement of the AdS/CFT
correspondence:
*
Zgrav [φi0 (x); ∂M ] =
exp −
XZ
!+
dd xφi0 (x)Oi (x)
i
(14.2)
CF T on ∂M
This is called the GKPW dictionary.∗ The index i runs over all the light fields in
the bulk effective field theory, and correspondingly over all the low-dimension local
operators in CFT.
The left-hand side
The lhs of (14.2) is the gravitational partition function in asymptotically AdS space.
It is formally computed by the same path integral that we discussed in the context of
black hole thermodynamics. Since AdS has a boundary, we must provide boundary
conditions to define this path integral. The boundary conditions on bulk scalars are
φi (z, x) = z d−∆ φi0 (x) + subleading as z → 0 .
(14.3)
where the mass of the bulk scalar is related to the scaling dimension of the CFT
operator by
2
m = ∆(d − ∆) ,
d
∆= +
2
∗
r
d2
+ m2 `2 .
4
(14.4)
After hep-th/9802109 by Gubser, Klebanov and Polyakov and hep-th/9802150 by Witten. I highly
recommend reading Witten’s paper.
131
We will see below that (14.3) is the leading solution of the wave equation for a bulk
scalar of mass m.
Similar statements apply to all bulk fields, including the metric, though the boundary
condition and formula for the dimension is slightly modified for fields with spin. The
boundary conditions on the metric involve a choice of topology as well as the actual
metric, which is why we’ve indicated explicitly that Zgrav depends on the boundary
manifold ∂M .
The right-hand side
The rhs of (14.2) is the generating functional of correlators in a CFT. In this equation
the φi0 (x) are sources, and the Oi (x) are CFT operators. Denoting the rhs of (14.2) by
Zcf t [φ0 ], correlation functions are computed in the usual way,
hO1 (x1 ) · · · On (xn )iCF T ∼
δn
i .
Z
[φ
]
cf
t
0
φ0 =0
δφ10 (x1 ) · · · δφn0 (xn )
(14.5)
The mapping
Each light field in gravity corresponds to a local operator in CFT. The spin of the
bulk field is equal to the spin of the CFT operator; the mass of the bulk field fixes the
scaling dimension of the CFT operator. Here are some examples:
Scalar: A bulk scalar field χ(z, x) is dual to a scalar operator in CFT. The boundary
value of χ acts as a source in CFT. This is exactly the relationship we used in our
derivation of the absorption cross section of the black string.
Graviton: Every theory of gravity has a massless spin-2 particle, the graviton gµν . This
is dual the stress tensor Tµν in CFT. This makes sense since every CFT has a stress
tensor. The fact that the graviton is massless corresponds to the fact that the CFT
stress tensor is conserved. It also fixes the scaling dimension to ∆T = d. We will see
this in more detail later.
Vector: If our theory of gravity has a spin-1 vector field Aµ , then the dual CFT has a
spin-1 operator Jµ . If Aµ is massless, then ∆J = d − 1 and Jµ is a conserved current.
Otherwise, ∆J > d − 1 and the current is not conserved.
132
This illustrated a general and important feature of AdS/CFT: gauge symmetries in the
bulk correspond to global symmetries in the CFT.
This is UV complete.
Note that CFTs are UV complete. Therefore (14.2) is a non-perturbative formulation of
a UV complete theory of quantum gravity. Shockingly, it is a definition of gravity from
a QFT without gravity. This is very powerful because we understand QFT relatively
well.
14.2
Example: IIB Strings and N = 4 Super-Yang-Mills
In some sense, it is believed that the AdS/CFT correspondence as summarized by
(14.2) holds for any theory of gravity and and CFT. That is, given a theory of gravity
we can use it to define a CFT via (14.2), and (perhaps) vice-versa. But aside from
certain examples, the correspondence is well defined and useful only in certain limits.
To illustrate this we turn to a specific example where AdS/CFT is understood in great
detail. This is the duality between IIB string theory and supersymmetric gauge theory:
IIB strings on AdS5 × S 5
=
Yang-Mills in 4d with N = 4 supersymmetry .
The gravity side
The string theory has adjustable scales ` ≡ `AdS , the Planck scale `P , and the string
scale `s . We do not need to use any details of string theory except to say that at
low energies, the effective action is Einstein + Matter + higher curvature corrections
suppressed by the string scale:∗
SIIB
1
∼
GN
Z
√
g R + Lmatter + `4s R4 + · · ·
(14.6)
The stringy states have masses of order 1/`2s , so at energies below 1/`2s it is just an
ordinary effective field theory like we discussed at the beginning of the course.
∗
There are no R2 corrections allowed with this amount of supersymmetry, but there are similar
examples with non-zero `2s R2 terms.
133
The CFT side
N = 4 Super-Yang-Mills is a highly supersymmetric gauge theory in 4d. Its matter
context is fixed uniquely by supersymmetry. It is just an SU (N ) gauge field plus all
the matter fields required by supersymmetry, which include matrix-valued scalar fields
transforming the adjoint representation of SU (N ) (unlike the fundamental representations we usually encounter in, say, QCD).
The gauge theory has two dimensionless parameters, N (ie the size of SU (N )) and the
Yang-Mills coupling constant gY M . Define the combination
λ = gY2 M N .
(14.7)
This is called the ‘t Hooft coupling. It turns out that gauge theory at large N is most
naturally organized as an expansion in λ and 1/N , rather than gY M and 1/N . This
is roughly because there are N fields running in loops, which changes the expansion
parameter from gY2 M to λ.
The mapping
The mapping from string theory parameters to CFT parameters is
λ∼
and
`d−1
AdS
∼
GN
`AdS
`string
`AdS
`P
4
d−1
(14.8)
∼ N2 .
(14.9)
(with known coefficients). We will see where this particular scaling comes from below in
more generality. For now we just want to note that this is a strong/weak duality: when
one side is easy, the other is (usually) hard. For example to have semiclassical Einstein
gravity, both loops and higher curvature corrections must be suppressed on the gravity
side. This means N 1 and λ 1 so the CFT is very strongly coupled. On the other
hand if we consider a weakly coupled CFT, then `s `AdS so stringy/higher curvature
corrections are not suppressed on the gravity side and this presumably behaves nothing
like ordinary gravity. (This is related to so-called ‘higher spin gravity’ or ‘Vasiliev
gravity’.)
134
14.3
General requirements
Returning to AdS/CFT in general, we can make some similar observations about when
it produces a nice semiclassical theory of gravity. This requires as least two things:
1. Strongly coupled CFT. If the CFT is weakly coupled, then there are too many
operators. For example, a free scalar field ψ leads to conserved currents of every
integer spin:∗
ψ∂µ ψ,
ψ∂µ ∂ν ψ,
ψ∂µ ∂ν ∂ρ ψ,
etc.
(14.10)
On the gravity side, this would require lots of massless or very light high-spin
states. This is something we expect in string theory at high enough energies but
not in our low energy effective field theory.
So we must require that the CFT has a sparse spectrum of low-dimension operators. This is sometimes called a large ‘gap’ in the spectrum, meaning a gap
between the low-energy fields and the stringy stuff. This can only happen at
strong coupling, although there can also be strongly coupled theories with no
gap which therefore do not have nice gravity duals.
2. Large Ndof . In the super-Yang-Mills example, we said GN ∼ 1/N 2 so that the
large number of degrees of freedom is required for gravity to be weakly coupled.
This is true in general, too. There are two ways to see this, both of which we
will discuss in more detail later. I will purposely be a little vague about the
definition of Ndof since there are several reasonable ways to define it, and they
are all different.
First, note that black hole entropy is S ∝ 1/GN , which is very large. Since
entropy is the log of the density of states, this means holographic CFTs must
have an enormous degeneracy of states at high energy. This means there are lots
of degrees of freedom. For example, a 2d CFT consisting of Nb free bosons has
√
S(E) ∝ Nb E.
Second, we can roughly measure the degrees of freedom by looking at the stresstensor 2pt function. This is fixed by conformal invariance up to a single coeffi∗
This is schematic, you must add corrections to these operators for them to be conserved by the
EOM.
135
cient:
hTµν (x)Tαβ (y)i = c × (known function of x, y) .
(14.11)
The coefficient c is a measure of degrees of freedom.∗ Consider again lots of free
fields: the stress tensors add, so the total stress tensor will have a very big 2pt
function.
On the gravity side, the stress tensor is dual to the graviton. We will see in detail
below how to calculate corelators, but for now suffice it to say that hT T icf t will be
related to a graviton scattering experiment hggigravity ∼ 1/GN . Thus c ∼ 1/GN
and we see again that weakly coupled gravity requires an enormous number of
degrees of freedom.
Clear there is a tension between requirements (1) and (2). We want lots of degrees
of freedom, and lots of states at high energies, but very few states at low energies.
Roughly speaking, you can think of this as the requirement that the CFT is confining:
it has lots of states at high energies, but very few at low energies where quarks are
confined. Later we will see a very direct link between black hole thermodynamics and
confinement.
14.4
The Holographic Principle
Many years ago Bekenstein conjectured that the maximum entropy you can fit into a
region of space is equal to the entropy of the corresponding black hole:
Smax =
area
.
4GN
(14.12)
This is called the Bekenstein bound. The argument is simple. If you have lots of stuff
in a region and Sstuf f > Sblackhole , then you can throw in some more stuff and form a
black hole. In doing so, the entropy of the system decreases! Therefore the second law
requires a bound like (14.12).†
This bound inspired ’t Hooft (in ’93) and later Susskind (in ’94) to argue that a theory
∗
But not an entirely satisfactory one. For example, it can increase under RG flow.
In the last few years this bound has been understood much better using entanglement entropy.
See for example 1404.5635 and references therein.
†
136
of quantum gravity must secretly live in fewer dimensions than our observed spacetime.
This principle is realized concretely by AdS/CFT.
137
15
Correlation Functions in AdS/CFT
As a first application of (14.2), we will use the gravity side to derive the correlators of
a conformal field theory. First we’ll start with a purely QFT discussion of correlators
in a theory with conformal invariance, then reproduce these results from gravity.
By the way, you’ve already seen one example of CFT correlators compute from gravity:
the absorption cross section calculation. That was related to a CFT correlator at finite
temperature. In this section are deriving correlators in the vacuum state, i.e., empty
AdS.
15.1
Vacuum correlation functions in CFT
This will be a brief introduction to CFT. For details, see: Polchinski’s String Theory
book; Kiritsis’s String Theory book; or the big yellow CFT book by Di Francesco et
al.
The group of conformal symmetries of Rd is SO(d + 1, 1). In Lorentz signature, the
conformal symmetries of Rd−1,1 are SO(d, 2). The generators of SO(d, 2) are
Pµ = −i∂µ
(15.1)
Lµν = −i(xµ ∂ν − xν ∂µ )
D = −ixµ ∂µ
Kµ = −i(2xµ xν ∂ν − x2 ∂µ )
The the first two lines are translations, rotations, and boosts; these generate the
Poincare group (which is 10-dimensional in d = 4). The 3rd line is the dilatation,
or scale generator, since under xµ → xµ + iDµ , the coordinate is just rescaled,
xµ → xµ (1 + ). The last line is called the special conformal transformation.
One way to derive (15.1) is to find the conformal Killing vectors of Minkowski space.
These are defined to be vectors V µ obeying
LV ηµν = f (x)ηµν ,
138
(15.2)
where f is any function. This is the infinitessimal version of the definition of a conformal
symmetry, which maps ds2 → eΩ(x) ds2 .
Operators in a CFT can be organized under representations of the conformal group.
We define primary operators to obey∗
[D, O(0)] = −i∆O(0)
[Kµ , O(0)] = 0 .
(15.3)
(15.4)
The dilatation eigenvalue ∆ is called the scaling dimension of O. The second condition
is like a highest weight condition. We can build the full representation by acting on
O(x) with the conformal generators, so for example ∂m uO(x) is a descendant operator.
The finite version of of the D commutator says that under a rescaling x → λx, we have
O(x) → λ∆ O(λx). More generally, primaries obey
0 µ −∆/d
∂x
O (x ) = det
O(x) .
ν
∂x 0
0
(15.5)
For a correlation function of n primaries this implies
hO1 (λx1 ) · · · On (λxn )i = λ−∆1 −∆2 −···−∆n hO1 (x1 ) · · · On (xn )i .
(15.6)
The special conformal transformations also impose requirements on correlators. It
turns out that all the conformal generators together completely fix the 2 and 3-point
functions of a CFT, up to overall factors. The two-point function of equal-weight fields
is
hO1 (x1 )O2 (x2 )i =
c12
|x1 − x2 |2∆
(∆1 = ∆2 ≡ ∆)
(15.7)
and it must vanish if ∆1 6= ∆2 . The number c12 can be rescaled by rescaling our normalization of the operators. Often we pick an orthonormal basis of primary operators,
so that cij = δij .
∗
This is for scalars. Operators with spin would also have the usual rule for action by the Lorentz
group, [Lµν , O(0)] = Σµν O(0).
139
Similarly, the only 3-point function allowed by conformal invariance is
hO1 (x1 )O2 (x2 )O3 (x3 )i =
|x12
|∆1 +∆2 −∆3 |x
23
c123
∆
| 2 +∆3 −∆1 |x
∆3 +∆1 −∆2
31 |
,
(15.8)
where
xij ≡ xi − xj .
(15.9)
The number cijk , called an OPE coefficient (for operator product expansion), is a
real physical prediction of the theory, since we’ve already fixed normalizations via the
2-point function.
In fact, the set of scaling dimensions ∆i and the OPE coefficients cijk are all the data
of a CFT. This is because higher correlators can be computed, at least in principle, by
sewing together 3-point functions and summing over intermediate states.
The 4-point function is not completely fixed by conformal symmetry, but it is highly
constrained. With equal external weights ∆1,2,3,4 = ∆, the most general form of the
4-point correlator is
hO(x1 )O(x2 )O(x3 )O(x4 )i = |x12 |−2∆ |x34 |−2∆ F (u, v)
(15.10)
where F is an arbitrary function of the conformal cross ratios,
x212 x234
u= 2 2 ,
x13 x24
15.2
x214 x223
v= 2 2 .
x13 x24
(15.11)
CFT Correlators from AdS Field Theory
We assume a strongly coupled, large-N CFT with a semiclassical holographic dual.
This is a limit where the gravity theory is weakly coupled, GN `, and higher curvature corrections can be neglected, `string ` (here ` is the AdS radius). According to
the GKPW dictionary (14.2), we can compute the generating function of CFT correlators on the gravity side by
Zcf t [φ0 ] ≡ he−
R
φ0 O
iCF T
`string
0
≈ exp −Sgrav + O(GN ) + O(
)
`AdS
140
(15.12)
(15.13)
where Sgrav is the on-shell action for gravity subject to the boundary condition
φ → z −∆+d φ0 (x)
(15.14)
as we approach the AdS boundary z → 0.
So to compute CFT correlators, we need to understand how to compute the classical
action in AdS as a functional of the boundary conditions.
This material is explained very clearly in many places, so I will not repeat it here. I
recommend reading Witten’s original paper on the subject, where ‘Witten diagrams’
were introduced [hep-th/9802150]. In class, I followed, almost exactly Kiritsis’s String
Theory in a Nutshell sections 13.8.1 and 13.8.2. Read those sections before continuing!
15.3
Quantum corrections
So far we only used the classical theory on the gravity side. (Though on the CFT side,
this is a strongly coupled QFT calculation which is not at all classical!) What happens
when we include loop corrections in the gravity? The gravitational loop expansion is
organized into powers of GN . The classical term is ∼ 1/GN . If we compute Witten
diagrams with loops, then we find an expansion in GN .
On the CFT side, this is an expansion in 1/Ndof , since recall the dictionary `d−1 /GN ∼
Ndof .
This implies something very special about CFTs with a semiclassical holographic dual:
These CFTs, although strongly coupled, have a meaningful expansion in 1/Ndof . Defining Ndof = N 2 (since this notation holds in SU (N ) gauge theory), this can be restated
as the fact that connected correlation functions are suppressed. That is, if we normalize
our operators by setting
hOOi ∼ 1
(15.15)
(with the appropriate factors of x suppressed), then the 3-point function is suppressed,
hOOOi ∼
141
1
N
(15.16)
and higher-point functions are dominated by their connected piece:
hOOOOi ∼ hOOihOOi + O(1/N 2 ) .
(15.17)
The explicit theories we know of with this sort of behavior are large-N gauge theories.
These have been studied for a long time, starting with a beautiful paper by ‘t Hooft
in the 70s where he showed that the Feynman diagrams of SU (N ) gauge theory in the
large-N limit naturally reorganize themselves into something that looks roughly like
a string theory. I will not cover this, but I highly recommend you read about it in
section 13.1 of Kiritsis, or the big AdS/CFT review [hep-th/9905111].
Another consequence of the weak coupling constant GN on the gravity side is that
gravity has an approximate Fock space. That is, if we have a weakly coupled scalar
field on the gravity side, then we can construct 1-particle states, 2-particles states, etc,
by acting with creation operators. On the CFT side, this means for example that if we
have a primary O1 of dimension ∆1 , and another primary O2 of dimension ∆2 , then
there is a third primary O1+2 of dimension
∆O1+2 ≈ ∆1 + ∆2 + O(1/N ) .
(15.18)
This is very special; it does not happen in general CFT, where states are just a some
strongly coupled mess and there is no way to ‘add’ some stuff to other stuff without
getting large corrections to the conserved charges from the strong interactions.
Following the gauge theory language, the operators dual to single bulk fields are called
‘single-trace operators’, and the operators like O1+2 are called ‘multi-trace operators’ and usually just denoted by the product O1 O2 (or more complicated things like
O1 n ∂µ1 ···µ` O2 ).
In words, (15.18) says that in a CFTs with a semiclassical holographic dual, low dimension operators have ‘small anomalous dimensions.’ I’ve restricted this statement
to low-dimension operators because these are the operators dual to bulk fields; high
dimension operators are dual to non-perturbative stuff like black hole microstates.
142
16
Black hole thermodynamics in AdS5
Now we return to black holes, and some of the techniques introduced in the beginning
of the course.
The basic starting point is that thermal states in CFT are dual to black holes in
quantum gravity. In fact, this is a special case of the dictionary (14.2), where we
impose boundary conditions appropriate for thermal field theory. That is,
Zcf t [φ0 ; M ] = Zgrav [φ0 ; boundary = M ]
(16.1)
where we take the manifold on which the CFT lives to be
M = Σd−1 × Sβ1 .
(16.2)
Here Σd−1 is space. We will mostly set Σd−1 = S`3 , a 3-sphere of size `. And Sβ1 is
a circle of size β. As we saw earlier in the course, the Euclidean path integral on
Σd−1 × Sβ1 defines the finite-temperature state on Σd−1 .
The meaning of the notation in (16.1) is that we calculate the gravity partition with
boundary condition φ0 on bulk fields, and boundary condition M on the bulk manifold.
Explicitly, fields obey the usual fall-off φ ∼ r−d+∆ φ0 (x) as r → ∞, and the metric itself
obeys the boundary condition
ds2 →
r2 2
`2 2
dt
dr + r2 dΩ23 ,
+
`2 E r2
tE ∼ tE + β .
(16.3)
Our goal is to compute the free energy at temperature β. For this we can turn off all
fields besides the metric, so φ0 = 0, and we just have the relation
Zcf t [β] = Zgrav [β] .
(16.4)
We will compute the rhs in gravity, and interpret it in CFT. The result will exhibit
rich behavior, including phase transitions as a function of temperature. This will turn
out to be related to confinement/deconfinement in the CFT.
143
16.1
Gravitational Free Energy
To compute Zgrav [β], in principle, we should compute the quantum gravity path integral
subject to the boundary condition (16.3). That’s impossible, but in the semiclassical
limit we can evaluate it approximately by expanding around classical solutions of the
equations of motion. We need to find all of the classical solutions that obey this
boundary condition, and evaluate their on-shell actions using
1
IE = −
16πGN
Z
12
d x g R+ 2 .
`
5
√
(16.5)
If there are several solutions, then the semiclassical approximation to the path integral
is
(1)
(2)
Zgrav (β) ≈ e−IE + e−IE + · · ·
(16.6)
Each saddlpoint also comes with an infinite series of perturbative (loop) corrections
but we won’t worry about those, we will just evaluate the classical contributions.
There are three classical solutions (in pure gravity) obey the thermal boundary condition (16.3): small black holes, large black holes, and thermal AdS.
16.1.1
Schwarzschild-AdS
The Euclidean black hole satisfying the boundary condition (16.3) is called SchwarzschildAdS, with metric
ds2 = f dt2E +
dr2
+ r2 dΩ23 ,
f
f =1+
r2
µ
− 2 ,
2
`
r
(16.7)
with the thermal identification
tE ∼ tE + β .
(16.8)
µ is a constant that will be related to the mass. The explicit metric on the unit 3-sphere
is
dΩ23 = dψ 2 + sin2 ψ(dθ2 + sin2 θdφ2 ) ,
144
(16.9)
where ψ and θ run from 0 to π and φ ∈ [0, 2π). The horizon is the outermost solution
of the equation f (r+ ) = 0, which gives
`2
2
r+
=
2
r
−1 +
4µ
1+ 2
`
!
.
(16.10)
As usual, the condition that this solution is non-singular at r = r+ relates β to r+ .
From the path integral point of view, this is because only smooth classical solutions
are good saddlepoints; solutions with a conical defect do not satisfy the equations of
motion at the defect. The conical defect trick gives
ie
2π`2 r+
β= 2
,
2r+ + `2
(16.11)
"
#
r
π`2
2β 2
r+ =
1± 1− 2 2 .
2β
π `
(16.12)
Note two things: first, there is a maximum β, ie minimum temperature,
`π
βmax = √ .
2
(16.13)
Second, for any given temperature β, there are two different black holes, corresponding
to the sign choice in (16.12). Call these the ‘small’ (minus sign) and ‘’large’ (plus sign)
black holes. The turnover is at
√
r∗ = `/ 2
(16.14)
so each β allows a small black hole with r+ < r∗ and a large black hole with r+ > r∗ .
We are working with thermal boundary conditions that fix the temperature β. So in
field theory language, we are working in the canonical ensemble. Therefore we should
sum over the allowed solutions; the thermodynamics will be determined by whichever
has the lower free energy. We will see below that the large black hole always has lower
free energy than the small black hole (but there is also a third solution, so the large
black hole is not always dominate).
The behavior (16.12) is quite different from flat spacetime. In flat spacetime, larger
black holes always have smaller temperature; this means ordinary Schwarzschild has
145
negative specific heat and so is thermodynamically unstable. (It cannot be held in
equilibrium with a bath, because it will absorb radiation from the bath and get colder
the more radiation it absorbs!)
On the other hand in AdS, according to (16.12), large black holes have positive specific
heat. If you make them bigger (higher energy), they get hotter. Small black holes
have negative specific heat, and very small black holes r+ ` don’t care about the
cosmological constant at all so are just like the flat spacetime Schwarzschild solution.
On-shell action
The free energy F = − β1 log Z is computed using the on-shell Einstein action. We must
be careful about the boundary terms. The full action we will use is
1
IE = −
16πGN
Z
√
12
1
d x g(R + 2 ) +
`
8πGN
5
Z
4
√
Z
d x γK +
r=1/
√
d4 x γLct [γ] .
r=1/
(16.15)
The first term is the usual Einstein term. The second term is the Gibbons-Hawking
boundary term. The last term is a counterterm; it can be any function of the intrinsic
boundary geometry γµν and will be picked to cancel divergences. Note that we are
cutting off the spacetime at r = 1/, we will take → 0 at the end.
Bulk term
The Einstein equation in empty spacetime implies R = −20/`2 . Thus the bulk term
in (16.15) is
Ibulk
1
=
2πGN `2
Z
√
d5 x g
(16.16)
r<1/
Z β
Z 1/
Z
1
3
=
dtE
drr
dΩ3
2πGN `2 0
r+
πβ
1
4
− r+ .
=
4GN `2 4
(16.17)
(16.18)
Boundary terms
To compute the Gibbons-Hawking-York boundary term we just plug into the definition
146
of extrinsic curvature, and eventually find
IGHY
1 3`2 µ`2
πβ
.
=−
−
GN `2 4 + 42
2
(16.19)
So far our answer Ibulk + IGHY is divergent as → 0. To fix this we need to add a
boundary counterterm which is a functional of γ. It turns out the only choice that
makes everything finite is
3
Ict =
8πGN `
`2
d x γ 1 + R[γ]
12
r=1/
Z
4
√
(16.20)
where R[γ] is the Ricci scalar for the metric γ.
Total
Plugging into the counterterm action, evaluating it, and adding everything up we find
(as → 0)
IE = Ibulk + IGHY
π2β
3`4
2 2
4
+ Ict =
.
r ` − r+ +
8GN `2 +
4
(16.21)
This is finite, by design. This should be viewed as a function of temperature IE = IE (β),
so we should plug in for r+ (β) using (16.12). If we pick the plus sign we get the action
of the large black hole, and if we pick the minus sign we get the action of the small
black hole. You can easily check that
large
small
IE (r+
(β)) ≥ IE (r+
(β)) .
(16.22)
Thus the dominant solution, with lower free energy, is the larger black hole, at any β.
Energy and entropy
Now that we have the partition function we can use all the usual thermodynamic
relations to derive things like energy and entropy. The energy is
3π 2
E = −∂β log Z =
8GN
`2
µ+
.
4
(16.23)
The first term is the relation between mass and µ. The second term is a little surprising,
since it is independent of the mass! It can be interpreted as a Casimir energy induced
147
by putting the theory on S 3 . Had we chosen boundary conditions R3 × Sβ1 , this term
would be zero.
The thermodynamic entropy is
3
π 2 r+
S = (1 − β∂β ) log Z =
2GN
(16.24)
which you can check agrees with the area law S = area/4GN .
16.1.2
Thermal AdS
We’re not done: there is a third solution with thermal boundary conditions (16.3). It
is called Euclidean thermal AdS and the metric is simply
2
ds =
r2
1+ 2
`
dt2E +
dr2
2
2
2 + r dΩ3 ,
1 + r`2
(16.25)
with the identification
tE ∼ tE + β .
(16.26)
This is just the metric of empty Euclidean AdS, except that we have identified the
Euclidean time circle. Note that β is not related to any parameter in the metric itself,
since there is no horizon r+ in this metric. Therefore β is not fixed by any regularity
condition, it is just a free parameter in this solution.
Lorentzian interpretation
We are discussing Euclidean manifolds, so let’s pause to comment on the Lorentzian
interpretation of all this. As discussed earlier in the course, path integrals of quantum
fields on a Euclidean manifold with tE ∼ tE +β prepare the fields in a thermal state. For
both of the Euclidean manifolds discussed here — the Euclidean black hole and thermal
AdS — the path integral on the Euclidean manifold prepares a thermal state on the
Lorentzian manifold. That is, the Euclidean path integral on Euclidean SchwarzschildAdS prepares the Hartle-Hawking thermal state for fields on the the Lorentzian black
hole. The Euclidean path integral on Euclidean thermal AdS prepares fields in a
thermal state on ordinary Lorentzian AdS.
148
So in other words: In Lorentzian signature, thermal AdS is exactly the same classical
solution as empty AdS, but the state of the perturbative fields is different — they are
thermally populated, but their energy is small O(~) and does not backreact on the
geometry itself.
Contractible vs non-contractible time circle
In the Euclidean black hole, tE is the angle in polar coordinates. Together, the coordinates (r, tE ) with r > r+ and tE ∈ (0, β) make a disk. The origin of the disk is a
smooth point which corresponds to the Euclidean horizon.
In thermal AdS, tE is a circle, but the circle does not contract anywhere. There is no
origin. So in this geometry the coordinates (r, tE ) make a cylinder rather than a disk.
Action
The calculation of the on-shell action is similar to what we did for the black hole.
Skipping the details, the answer in the end is just the Casimir term:
(th)
IE
π2β
=
8GN `2
3`4
4
.
(16.27)
This is the free energy, which we can use to calculate the energy and entropy.
16.1.3
Hawking-Page phase transition
We found three Euclidean geometries obeying the thermal boundary condition (16.3).
They are the small black hole, large black hole, and thermal AdS. The free energy of
the large black hole is always smaller than that of the small black, so in understanding
the phases, we can forget about the small black hole – it never dominates the canonical
ensemble.
This leaves the large black hole and thermal AdS with actions
3`4
π2β
2 2
4
r ` − r+ +
=
8GN `2 +
4
4
2
π β
3`
(th)
IE (β) =
,
2
8GN `
4
(bh)
IE (β)
149
(16.28)
(bh)
where in IE
we choose the larger root for r+ (β).
The semiclassical approximation to the gravitational path integral is the sum
(bh)
(th)
Zgrav (β) ≈ exp(−IE ) + exp(−IE ) + . . . .
(16.29)
Each of the exponents is very large, since they are order 1/GN . Therefore the sum is
exponentially dominated by whichever term is bigger:
(bh)
log Zgrav (β) ≈ max −IE ,
(th)
−IE
.
(16.30)
There is a sharp (1st order) phase transition where the two solutions exchange dom(th)
inance, ie at IE
(bh)
= IE . Comparing the two actions, the critical temperature, and
corresponding black hole radius, for this phase transition is
βcrit =
2π`
,
3
crit
r+
=`.
(16.31)
The low-temperature phase is thermal AdS; the high temperature phase is the black
hole. This phase transition is called the Hawking-Page transition and was discovered
well before AdS/CFT. The story is qualitatively the same in any number of dimensions,
AdSd+1 (with a few differences in AdS3 ).
Entropy
The entropy of thermal AdS is zero. We can see this either by noting there is no
horizon, or computing (1 − β∂β ) log Z = 0. Actually, it is not exactly zero, since we
have only computed the semiclassical term. There are quantum corrections, and the
true entropy is O(G0N ) from the one-loop contribution (i.e., determinant of gravitons
matter fields in AdS).
Thus full thermal entropy S(β), accounting for the phase transition, is O(G0N ) at low
temperatures and then suddenly jumps to a very large number O(1/GN ) at βcrit . In
the microcanonical ensemble, where we view this as a function of energy S(E), the
entropy is related to the density of states
S(E) = log ρ(E) .
150
(16.32)
The Hawking-Page transition indicates that theories with a semiclassical gravity description must have a small number of states at low energy, but an enormous number
of states at high energy, with a sharp transition.
16.1.4
Large volume limit
We have been computing the free energy at temperature β for the theory on the space
S`3 , a 3-sphere of size `. In fact since we are in conformal field theory, only the ratio
`/β is meaningful, as this is the only dimensionful parameter. In other words the only
parameter is `T . Going to high temperatures is therefore the same as going to large `.
If we are interested in the theory on R3 we can take ` → ∞. This is the same as taking
the temperature T → ∞. In this limit, from (16.28), the free energy becomes (with
IE = βF )
F ≈−
`2
GN
`3 π 6 T 4 ,
(16.33)
i.e.,
F ∼−
`
`P
3
V T4
(16.34)
where V is the volume of the system. Up to the prefactor, we could have guessed
this answer from dimensional analysis. In a conformal field theory on Rd−1 the only
dimensionful scale is the temperature, and F must be proportional to volume, so
conformal invariance implies F ∼ V T d .
Note that the theory on the plane has only one phase: the black hole phase. There
is no Hawking-Page transition on R3 . It is essentially always in the high temperature
phase.
16.2
Confinement in CFT
Any CFT with a semiclassical holographic dual must share the same thermodynamics,
summarized by (16.30). What does this mean about the CFT? The microcanonical
entropy tells us about the spectrum: we must have an enormous number of degrees of
freedom to reproduce the high-energy density of states. However we must have a small
151
number of states at low energies. This sounds like confinement! In a confining SU (N )
gauge theory, in the confining phase, the physical states are color singlet hadrons, and
the free energy is F = O(1). In a deconfined phase, the states are gluons, and the free
energy is F ∼ O(N 2 ). This agrees with a our results above (after subtracting off the
contribution from the Casimir energy, which is a temperature-independent contribution
to F and does not affect the entropy). The black hole phase is like the deconfined phase,
and the confined phase is like thermal AdS.
This analogy comes with some caveats, so let’s compare and contrast QCD with
a holographic theory like N = 4 Super-Yang-Mills. In QCD, there is a confinement/deconfinement phase transition in infinite volume, ie for the theory on R3 . It is
confining at low temperatures and deconfined at high temperatures. According to our
gravity results, N = 4 Super-Yang-Mills (at strong coupling) does not have a confining
phase on R3 . CFT’s on R3 cannot have phase transitions, because the temperature
can always be rescaled (unless there is some other parameter turned on, like a chemical
potential). So in fact N = 4 SYM is not a confining gauge theory in the same sense
as QCD.
In gravity, the phase transition is on S 3 . Normally it is not possible to have a phase
transition in finite volume – with a finite number of degrees of freedom, the free energy
is an analytic function of β, and we get sharp phase transitions only in the thermodynamic limit. However this is possible in gravity because of the large-N limit. The
same statements are true of N = 4 SYM on a sphere: it has something like a confinement/deconfinement transition on the sphere, in the N → ∞ limit. It is not the same
sort of confinement as QCD, which comes from dynamics in a very complicated way.
In gauge theory on the sphere, we get ‘kinematic confinement’ just from the Gauss
law constraint, which does not allow charges states on a compact space. Therefore
the physical states on a sphere cannot have any net color. This is what ‘confines’ the
theory so that there are O(1) physical states at low energies.
Temporal Wilson loop
(This will be very brief; see Kirtsis for more discussion.)
H
Define the temporal Wilson loop W = Tr exp A where the integral is over a worldline
going around the thermal time circle. This is an order parameter for the deconfinement
152
transition:
hW i 6= 0
⇒
deconfinement
hW i = 0
⇒
confinement
⇒
⇒
black hole phase
(16.35)
thermal AdS phase
(16.36)
(hW i =
6 0 actually breaks a symmetry: the center of the gauge group, ZN ∈ SU (N ).)
A rough explanation is that you can think of the temporal Wilson loop as a free quark.
If a free quark has finite energy, then you get hW i =
6 0, but if the free quark has infinite
energy then hW i = 0.
To compute a Wilson loop in AdS/CFT from the gravity side, the rule is to find a
string worldsheet ending on the Wilson line and extending into the bulk. This classical
string diagram computes the leading contribution to the Wilson loop at large N .
In the Euclidean black hole, since (tE , r) make a disk, it is easy to find a string worldsheet ending on this Wilson line. In thermal AdS, however, since the thermal circle is
not contractible — ie (tE , r) make a cylinder — you cannot find such a string worldsheet, and the Wilson line vanishes.
Thus the deconfined phase is the phase with a contractible thermal circle in the dual
geometry, and the confined phase has a non-contractible thermal circle.
16.3
Free energy at weak and strong coupling
So can we calculate the free energy of N = 4 SYM, and compare to (16.30)? Unfortunately, no. The gravity calculation is dual to N = 4 SYM at very strong ‘t Hooft
coupling, λ ≡ gY2 M N → ∞. The free energy is not protected by supersymmetry, and
it is unknown how to do this calculate in gauge theory at strong coupling.
But we can do the CFT calculation at weak coupling. The free energy of a weakly
coupled QFT is just a 1-loop calculation, ie a determinant for each of the fields. This
calculation has been done. It agrees qualitatively, but not quantitatively, with the
gravity calculation. For example, after translating all the parameters of CFT to gravity
153
parameters, the free energy of free N = 4 SYM on R3 is
4
Ff ree = Fgravity ,
3
(16.37)
where Fgravity is given in (16.33). This famous factor of 4/3 is not a contradiction. It
just means that the free energy at strong coupling is different from the free energy at
weak coupling.
In principle, or perhaps on a lattice, the free energy is some function of the coupling,
F = −f (λ)
π2 2
N V T4 .
6
(16.38)
We know the behavior of f (λ) as λ → 0 and as λ → ∞. We only described the
leading terms above, but it is also possible to calculate corrections. On the CFT
side, corrections come from higher loops. On the gravity side, corrections come from
including higher curvature (stringy) contributions to the classical action. (In principle
we could also ask about 1/N corrections, which would require quantum calculations on
the gravity side, but we’ll restrict to the leading-N behavior.) These corrections have
been calculated and lead to
gravity: f (λ) =
3 45 ζ(3)
+
+ ···
4 32 λ3/2
as λ → ∞
(16.39)
and
CFT: f (λ) = 1 −
3
λ + ···
2π 2
as λ → 0
(16.40)
Evidently the corrections are heading the right direction, but the full function f (λ) is
unknown.
Exercise: Hawking-Page in Three Dimensions
Recall the metric of the Euclidean BTZ black hole in AdS3 ,
2
2
ds = `
2
(r −
8M )dt2E
dr2
+ 2
+ r2 dφ2
r − 8M
154
.
(16.41)
In a previous exercise, you computed the on-shell Euclidean action of this black hole.
The answer, including all boundary terms and counterterms, is
(bh)
SE (β) = −
π2c
3β
(16.42)
where
c=
3`
.
2GN
(16.43)
1. Like in AdS5 , there is also a thermal AdS solution with the same boundary
condition.∗ We will use a trick to compute its action. The trick is to note
1
that (16.41) is a solid torus with boundary S2π`
× Sβ1 . (The subscript is the
circumference of the S 1 ). The thermal circle Sβ1 is ‘filled in’ to make the solid
torus.
1
Thermal AdS3 is a solid torus where instead the other circle S2π`
is ‘filled in’ to
make solid torus. Argue that this implies
(th)
(bh)
SE (β) = SE (
4π 2 `2
cβ
)=−
.
β
12
(16.44)
Comment: This is a special case of a modular transformation. It is a ‘large’†
conformal transformation acting on a torus, which roughly speaking relates a fat
torus to a skinny torus.
2. Sketch a plot of the free energies F (bh) and F (th) . Find the critical temperature
βcrit of the Hawking-Page phase transition, and write log Z(β) as a piecewise
function.
3. Find the thermodynamic entropy S(β) for all β > 0.
4. Find the energy E(β) for all β > 0.
5. Use part (4) in part (3) to find the entropy in the microcanonical ensemble,
S(E). (Be careful about what ranges of E your formulas apply to; in particular
you cannot find S(E) for all E by this method.)
∗
†
Unlike AdS5 , there is only one black hole with temperature β.
i.e., not continuously connected to the identity
155
6. Interpret your results in terms of the density of states in a 2d CFT dual to 3d
gravity.
156
17
Eternal Black Holes and Entanglement
References: This section is based mostly on Maldacena hep-th/0106112; see also the
relevant section of Harlow’s review lectures, 1409.1231.
An eternal black hole is the black hole with the full, two-sided Penrose diagram. It
has a past singularity, a future singularity, and two asymptotic regions:
(17.1)
This is to be distinguished from a black hole that forms from gravitational collapse,
which has no past singularity and no second asymptotic region on the ‘left’ of the
Penrose diagram. Although we often use the maximally extended Penrose diagram to
discuss all sorts of black holes, it is only in the eternal black hole that we should really
take the left side of the Penrose diagram seriously.
An eternal black hole in AdS — the maximally extended AdS-Schwarzschild spacetime
— has two boundaries. This means that it is dual to two copies of the CFT. In fact,
the connection between thermal field theory and this ‘doubling’ of degrees of freedom
was well known long ago, and is called the thermofield double formalism. First we will
describe this formalism in QFT, then we’ll make the connection to AdS black holes.
17.1
Thermofield double formalism
Consider any QFT, with Hamiltonian H and complete set of eigenstate |ni,
H|ni = En |ni .
157
(17.2)
The thermofield double formalism is a trick to treat the thermal, mixed state ρ = e−βH
as a pure state in a bigger system. First we double the degrees of freedom, i.e., we
consider a new QFT which is two copies of the original QFT. If the theory is defined
by a Lagrangian, then for every field φ in the original QFT, there are two fields φ1 (x1 )
and φ2 (x2 ) in the doubled QFT. These two fields live in different spacetimes x1 and
x2 , and are not coupled in the Lagrangian at all. The states of the doubled QFT are
|mi1 |ni2 .
(17.3)
Now in this doubled system we consider the thermofield double state:
X
1
e−βEn /2 |ni1 |ni2 .
|T F Di = p
Z(β) n
(17.4)
This is a particular pure state in the doubled system. The density matrix of the doubled
QFT in this state is
ρtotal = |T F DihT F D| .
(17.5)
The reduced density matrix of system 1 is
ρ1 = tr2 ρtotal
!
X
X
e−βEn /2 |ni1 |ni2 2 hn0 | 2 hn0 |e−βEn0 /2 |mi2
=
2 hm|
n,n0
m
=
X
= e
e−βEn |ni1 1 hn|
n
−βH1
(17.6)
Therefore, if we restrict our attention to system 1, this pure state in the doubled system
is indistinguishable from a thermal state. For example, if O1 is made of local operators
acting on system 1, O1 = φ1 (x1 )χ1 (y1 ) · · · , then
hT F D|O1 |T F Di =
1
Tr
Z(β)
H1
e−βH1 O1 .
(17.7)
This procedure is called purifying the thermal state. In fact, any mixed state can be
purified by adding enough auxiliary states and tracing them out.
Although systems 1 and 2 are not coupled in the Lagrangian of the doubled system,
158
they are correlated because we are in this particular entangled state. For example, if
O1 is built from operators acting on system 1 and O2 is built from operators acting on
system 2, then
hT F D|O1 O2 |T F Di
(17.8)
can be non-zero.
The Hamiltonian
The choice of Hamiltonian acting on the doubled system is up to us. Two convenient
choices are
Htot = H1 − H2
and
H̃tot = H1 + H2 .
(17.9)
For our purposes, we will just use Htot , but H̃tot is also useful in other contexts. Under
Htot , the TFD state is time-independent, since the phases cancel:
|T F D(t)i ≡ e−iHtot |T F Di =
X
e−βEn /2 e−i(H1 −H2 ) |ni1 |ni2 = |T F Di .
(17.10)
n
17.2
Holographic dual of the eternal black hole
The statement
Maldacena’s proposal is that the eternal black hole depicted in (17.1) is dual to two
copies of the CFT, in the thermofield double state |T F Di. Each asymptotic boundary
of AdS is a copy of the original dual CFT. So, for example, to compute correlation
functions like
hT F D|φ1 (x1 )χ2 (x2 )|T F Di
(17.11)
we would use Witten diagrams with χ inserted on the left boundary, and φ inserted on
the right boundary. Note that the local bulk fields are not doubled: there is just one
bulk field Φ dual to the boundary operators φ1 and φ2 , but this makes sense because
we have to specify double the boundary conditions for Φ. The boundary condition for
Φ on the left acts like a source for φ2 , and the boundary condition for Φ on the right
acts like a source for φ1 .
159
The Hamiltonian
The Hamiltonian Htot in (17.9) has a natural bulk interpretation. It is dual to the
bulk Hamiltonian that generates time evolution along the isometry ∂t , where t is the
usual Schwarzschild coordinate. Recall (or look back at a textbook on the Kruskal
coordinate change) that the Schwarzschild t coordinate runs ‘backwards’ on the left
side of the Penrose diagram. That is, all of the spatial slices drawn in this figure are
equivalent under the ∂t isometry:
(17.12)
This corresponds to the minus sign in Htot = H1 − H2 .
Derivation
To justify the claim that the eternal black hole is dual to the TFD state, we will apply
the AdS/CFT dictionary (14.2), in the form
Zgravity [∂M = Σ] = Zcf t [Σ] .
(17.13)
(Here M is the bulk manifold, and the meaning of the lhs is the gravity path integral
with boundary condition ∂M = Σ.)
First, the CFT: The Euclidean path integral that prepares the TFD state is a path
integral on an interval of length β/2, times a circle:
Σ = Intervalβ/2 × S d−1 .
160
(17.14)
Pictorially,
(17.15)
This path integral has two open cuts (red), at the ends of the interval. We interpret the
left cut as defining a state in system 2, and the right cut as defining a state in system
1. That is, this picture should be interpreted as a rule for computing the transition
amplitude with field data ϕ1 and ϕ2 specified at the ends of the interval. To confirm
that this path integral really prepares the TFD state, all we need to do is check that
it computes the correct transition amplitudes. The path integral with these boundary
conditions is∗
1 hϕ1 | 2 hϕ2 |T F Di
= hϕ1 |e−βH/2 |ϕ∗2 i
X
=
ϕ1 |nihn|ϕ˜2 ie−βEn /2
(17.16)
(17.17)
n
=
X
e−βEn /2 hϕ1 |ni1 hϕ2 |ni2
(17.18)
n
These are precisely the matrix elements of the state |T F Di defined in (17.4). So, as
claimed, this is the Euclidean path integral that prepares |T F Di.
Now that we’ve produced this state from a Euclidean path integral on the manifold
Σ, we can apply (17.13). We must find a Euclidean gravity solution with conformal
boundary condition ∂M = Intervalβ/2 × S d−1 . In fact, half of the Euclidean black hole
has precisely this boundary condition. That is, we consider the Euclidean ScharzschildAdS solution and restrict to tE ∈ [0, β/2] instead of the full range tE ∈ [0, β]. The
(tE , r) portion of this Eulidean spacetime makes a half-disk; the boundary of the halfdisk is Intervalβ/2 × S d−1 . The half-disk is cut down the middle; this cut is interpreted
as the time=0 surface of the Lorentzian spacetime. Pictorially, the bulk spacetime has
a Euclidean piece that prepares the state, then a Lorentzian piece describing the time
∗
The tildes indicate the conjugate state.
161
evolution in Minkowski signature:
(17.19)
The red blobs in this picture denote the S d−1 ’s at the end of the interval on the
boundary, the red circles in (17.15).
17.3
ER=EPR
Let’s describe this result in words. The left side of the Penrose diagram is dual to
CF T2 , and the right side is dual to CF T1 . The Einstein-Rosen bridge connecting
to the two sides, the black hole interior itself, are somehow ‘created’ by entangling
CF T1 with CF T2 . In fact this is a precise statement: In CFT language, correlators
between the two CFTs like hT F D|O1 O2 |T F Di are non-zero only because we are in
the entangled state |T F Di. After all, the two CFTs are not coupled. In the bulk,
these correlations are nonzero because we can draw Witten diagrams going through
the interior. For very massive fields or high energies, the left-right correlation functions
can be approximated by geodesics that pass through the black hole interior. Without
a wormhole connecting the two sides, there would be no such correlations.
This idea and its generalizations have recently been given the slogan ‘ER=EPR’ (by
Maldacena and Susskind): Einstein-Rosen bridges are equivalent to entanglement (as
discussed by Einstein-Podolsky-Rosen). This slogan is only entirely precise and well
defined in the semiclassical limit, describing the eternal black hole and similar spacetimes, but the idea is that some more general construction should make sense in the
very quantum, non-geometrical limit.
162
17.4
Comments in information loss in AdS/CFT
Hawking’s information loss paradox relied on a black hole that forms from collapse,
then evaporates. In AdS, this only happens for small black holes. These black holes
are not in thermal equilibrium, and are difficult to address precisely using AdS/CFT.
Of course, the CFT is always unitary, so if we believe AdS/CFT (or use AdS/CFT to
define a theory of quantum gravity) then obviously this evaporation process, however
it is described in CFT, must be unitary. This strongly suggests that unitarity should
be preserved, and locality or some other tenet of effective field theory must be violated.
However it is not very satisfying, since it does not answer the question of what went
wrong with Hawking’s calculation. Presumably the answer is that local effective field
theory is not quite right in non-perturbative quantum gravity, but we do not really
understand how to characterize this breakdown. This is a very important open question
in current research.
17.5
Maldacena’s information paradox
Maldacena introduced a different version of the information paradox that applies to
large, eternal black holes. This version is easier to address in AdS/CFT. The idea is
to first perturb the thermal state by inserting an operator O2 in CF T2 ,
|T F Di → |T]
F Di = (1 + O2 )|T F Di .
(17.20)
This changes the reduced density matrix of system 1,
ρ1 → ρ̃1 = e−βH1 + tiny corrections .
(17.21)
Now, we compute expectation values in CFT1 ,
hT]
F D|O1 |T]
F Di
(17.22)
in the perturbed state. To first order in the perturbation, this is the two-sided correlation function
hO1 i ∼ G12 ≡ hT F D|O1 O2 |T F Di .
163
(17.23)
Now we can produce a contradiction by waiting a very long time, so this correlation
function decays. On the gravity side, if we hold O2 at a particular time and send O1
to very late times, then the geodesic distance between these two points grows linearly
with time, forever. Therefore the correlation function must decay as
Ggravity
∼ e−const×t/β
12
(17.24)
for t β. This decays exponentially to zero. At very late times, it therefore becomes
exactly thermal, with arbitrarily small corrections.
This contradicts unitarity of the CFT. In the CFT, any perturbation of the thermal
state should stay forever a perturbation of the thermal state: it will of course become
scrambled and appear to thermalize, but it should never forget the initial perturbation
completely, so it should never become arbitrarily close to the thermal state. In fact
the corrections to the thermal state should be suppressed by the entropy, but finite:
T
GCF
∼ e−const×S
12
(17.25)
for t β. In summary, at very late times, gravity ‘forget’ the initial perturbation, but
a unitary CFT does not:
Ggravity
Gunitary
.
12
12
(17.26)
However is this paradox resolved? The answer is that we have neglected non-perturbative
contributions of the gravity side of order e−1/GN ∼ e−S . For example, there is another
saddlepoint (the thermal AdS saddle) and fluctuations around this saddle will also
contribution to the two-sided correlation function at this order.
Although this tells us where the gravity derivation went wrong, it does not tell us
exactly how to recover the lost information in quantum gravity, i.e., without referring
to the dual CFT. Presumably this would require treating the full non-perturbative
string theory, which is currently not possible.
164
17.6
Entropy in the thermofield double
Our next topic will be entanglement. It will be a while before we get back to gravity,
so as a brief preview, let’s consider the interpretation of entropy in the thermofield
double formulation of the black hole. The state |T F Di is a pure state; pure states
have no ordinary entropy, i.e.,
ρtotal ≡ |T F DihT F D|
(17.27)
Stot = − tr ρtotal log ρtotal = 0 .
(17.28)
has entropy
However, if we trace out half the system, we know this gives a thermal state, so it
should have some entropy. The reduced density matrix of system 1 is
ρ1 = tr2 ρtotal = e−βH1
(17.29)
Therefore the state of system 1 has entropy,
S1 = − tr ρ1 log ρ1 = Sthermal (β) 6= 0 .
(17.30)
We started in a pure state, with no entropy. Where did the entropy of system 1
come from? The answer is entanglement. Thermal entropy is just one special case of
entanglement entropy.
165
18
Introduction to Entanglement Entropy
The next few lectures are on entanglement entropy in quantum mechanics, in quantum
field theory, and finally in quantum gravity. Here’s a brief preview: Entanglement
entropy is a measure of how quantum information is stored in a quantum state. With
some care, it can be defined in quantum field theory, and although it is difficult to
calculate, it can be used to gain insight into fundamental questions like the nature of
the renormalization group. In holographic systems, entanglement entropy is encoded
in geometric features of the bulk geometry.
We will start at the beginning with discrete quantum systems and work our way up to
quantum gravity.
References: Harlow’s lectures on quantum information in quantum gravity, available on
the arxiv, may be useful. See also Nielsen and Chuang’s introductory book on quantum
information for derivations of various statements about matrices, traces, positivity, etc.
18.1
Definition and Basics
A bipartite system is a system with Hilbert space equal to the direct product of two
factors,
HAB = HA ⊗ HB .
(18.1)
Starting with a general (pure or mixed) state of the full system ρ, the reduced density
matrix of a subsystem is defined by the partial trace,
ρA = trB ρ
(18.2)
and the entanglement entropy is the von Neumann entropy of the reduced density
matrix,
SA ≡ − tr ρA log ρA .
(18.3)
Example: 2 qubit system
If each subsystem A or B is a single qubit, then the Hilbert space of the full system is
166
spanned by
|00i,
|01i,
|10i,
|11i ,
(18.4)
where the first bit refers to A and the second bit to B, i.e., we use the shorthand
|iji ≡ |iiA |jiB ≡ |iiA ⊗ |jiB .
(18.5)
Suppose the system is in the pure state
1
|ψi = √ (|00i + |11i) ,
2
(18.6)
so ρ = |ψihψ|. As a 4x4 matrix, ρ has diagonal and off-diagonal elements. Diagonal density matrices are just classical probability distributions, but the off-diagonal
elements indicate entanglement and are intrinsically quantum.
The reduced density matrix of system A is
ρA = trB ρ
1
=
B h0| (|00i + |11i) (h11| + h00|) |0iB
2
1
+ B h1| (|00i + |11i) (h11| + h00|) |1iB
2
1
=
|0iA A h0| + |1iA A h1|
2
∝ 12x2 .
(18.7)
The last line says ρA is proportional to the identity matrix of a 2-state system. In this
case we say ρA is maximally mixed, and the initial state |ψi is maximally entangled.
The entanglement entropy of subsystem A is easy to calculate for a diagonal matrix,
SA = − tr ρA log ρA
1
1
= −2 × log
4
4
= log 2 .
(18.8)
Interpretation of entanglement entropy
In fact the 2-qubit example illustrates a useful way to put entanglement entropy into
167
words:
Entanglement entropy counts the number of entangled bits between A and B.
If we had k qubits in system A and k qubits in system B, then in a maximally entangled
state SA = k log 2. So SA counts the number of bits, or equivalently, eSA counts the
number of entangled states (since k qubits have 2k states).
Rephrased slightly:
Given a state ρA with entanglement entropy SA , the quantity eSA is the minimal number
of auxiliary states that we would need to entangle with A in order to obtain ρA from a
pure state of the enlarged system.
Schmidt decomposition
A very useful tool is the following theorem, called the Schmidt decomposition: Suppose
we have a system AB in a pure state |ψi. Then there exist orthonormal states |iiA of
A and |iiB of B such that
|ψi =
X
λi |iiA |ĩiB ,
(18.9)
i
with λi real numbers in the range [0, 1] satisfying
X
λ2i = 1 .
(18.10)
i
The number of terms in the sum is (at most) the dimension of the smaller Hilbert space
HA or HB .
Proof: See Wikipedia, or Nielsen and Chuang chapter 2.
If A is small and B is big, this is intuitive. It says we can pick a basis for |iiA , and each
of these states will be correlated with a particular state of system B. The thermofield
double is an obvious example.
Complement subsystems
An immediate consequence of the Schmidt decomposition is that a pure state of system
168
AB has
SA = SB
(pure states) .
(18.11)
To see this, write the reduced density matrices in the Schmidt basis,
ρA =
X
λ2i |iiA A hi| ,
ρB =
i
X
λ2i |iiB B hi| .
(18.12)
i
Both density matrices have eigenvalues λ2i so they have the same entropy. (18.11) does
not hold for mixed states of AB.
18.2
Geometric entanglement entropy
Entanglement entropy can be defined whenever the Hilbert space splits into two factors.
A very important example is when we define A as a subregion of space.
Example: N spins on a lattice in 1+1 dimensions
Let’s arrange N spins in a line. Define A to be a spatial region containing k spins, and
B = AC is everything else:
The most general state of this system is
|ψi =
X
cs1 ···sN |s1 i|s2 i · · · |sN i
(18.13)
{si }
where si = 0 or 1 (meaning ‘up’ or ‘down’), and the c’s are complex numbers.
Scaling with system size
Let’s restrict to 1 |A| |B|, so that we can think of subsystem A as large and B
as infinite. In a random state, i.e., one in which the coefficients cij··· are drawn from
a uniform distribution, we expect any subsystem A to be almost maximally entangled
with B. In the language of the Schmidt decomposition, this means that λi is nonzero
169
√
and ∼ 1/ 2k for a complete basis of states |iiA . In fact this is a theorem, see Harlow’s
lectures for the exact statement.
Accordingly, the entanglement entropy scales as the number of spins in region A. In
1+1d this is linear in the size of A, and more generally,
SA ∼ Volume(A)
(random state).
(18.14)
In other words, most states in the Hilbert space of the full system have entanglement
scaling with volume.
However, often we are interested in the groundstate. Ground states of a local Hamiltonian are very non-generic, and the corresponding entanglement entropies obey special
scaling laws. Usually, if the system is gapped (i.e., correlations die off exponentially),
the ground state must obey the area law :
SA ∼ Area(A)
(ground state of local, gapped Hamiltonian) .
(18.15)
(This is a theorem in 1+1d, and usually true in higher dimensions.)
Thus groundstates occupy a tiny, special corner of the Hilbert space. This is a corner
with especially low ‘complexity.’ Intuitively speaking, a large degree of entnaglement
is what makes quantum information exponentially more powerful than classical information; so states with lower entanglement entropy are less complex. More specifically,
this actually means that you can encode a groundstate wavefunction with far fewer
parameters than the 2N complex numbers appearing in (18.13).
DMRG
In 1+1d, the area law becomes simply
SA ∼ const
(18.16)
independent of the system size. This special feature is responsible for a hugely important technique in quantum condensed matter called the density matrix renormalization
group (DMRG). This technique is used to efficiently compute groundstate wavefunc-
170
tions of 1+1d systems using a computer. This would not be possible for general states,
since (we think) classical computers require exponential time to simulate quantum systems. But (18.16) means that, in a precise sense, groundstates of gapped 1d systems
are no more complex than classical systems.
Scaling at a critical point
The area law applies to gapped systems. Near a critical point, where dof become massless and long-distance correlations are power-law instead of exponentially suppressed,
the area law can be violated. In a 1+1d critical system, and therefore also in 1+1d
conformal field theory, (18.16) is replaced by
SA ∼ log LA
(18.17)
where LA is the size of region A. This is bigger than the area law, but still much lower
than the volume-scaling of a random state.
18.3
Entropy Inequalities
Relative entropy
Much of the recent progress in QFT based on entanglement comes from a few inequalities obeyed by entanglement entropy. Define the relative entropy
S(ρ||σ) ≡ tr ρ log ρ − tr ρ log σ .
(18.18)
(Note that this is not symmetric in ρ, σ.) This obeys
S(ρ||σ) ≥ 0
(18.19)
with equality if and only if ρ = σ. The proof of this statement is straightforward,
see Wikipedia. It just involves some matrix manipulations. The key ingredient is the
fact that density matrices in quantum mechanics are very special: they have a positive
spectral decomposition,
ρ=
X
pi vi vi∗
i
171
(18.20)
where pi is non-negative and vi is a basis vector. This is necessary for quantum mechanics to have a sensible probabilistic interpretation and is closely related to unitarity.
The relative entropy can be viewed as a measure of how ‘distinguishable’ ρ and σ
are. In the classical case (diagonal ρ and σ), it is error we will make in predicting
the uncertainty of a random process if we think the probability distribution is σ, but
actually it is ρ. Given this interpretation, positivity is obvious — clearly we will never
do better using the wrong distribution.
Triangle inequality
Positivity or relative entropy implies the triangle inequality,
|SA − SB | ≤ SAB .
(18.21)
I(A, B) ≡ SA + SB − SAB .
(18.22)
Mutual information
Define the mutual information,
This can be written as a relative entropy, and is therefore non-negative:
I(A, B) = S(ρAB ||ρA ⊗ ρB ) ≥ 0 .
(18.23)
Roughly, I(A, B) measures the amount of information that A has about B (or viceversa, since it is symmetric).
In a pure state of AB, the only correlations between A and B come from entanglement,
so in this case I(A, B) measures entanglement between A and B. However, in a mixed
state, I(A, B) also gets classical contributions. For example in a 2-qubit system, it is
easy to check that the classical mixed state
ρAB ∝ |00ih00| + |11ih11|
has non-zero mutual information.
172
(18.24)
Strong subadditivity
So far we have discussed partitioning a system into two pieces A and B, but we can
partition further and find new inequalities. The strong subadditivity inequality (SSA
for short) applied to a tripartite system HABC = HA ⊗ HB ⊗ HC , is
SABC + SB ≤ SAB + SBC .
(18.25)
This is less mysterious if written in terms of the mutual information,
I(A, B) ≤ I(A, BC) .
(18.26)
Although this inequality seems obvious — clearly A has more information about BC
than about B alone — and is ‘just’ a feature of positive matrices, it is surprisingly
difficult to prove. See Nielsen and Chuang for a totally unenlightening derivation.
Sometimes it is useful to express (18.25) in different notation, where A and B are two
overlapping subsystems, which are not independent:
SA∪B + SA∩B ≤ SA + SB .
(18.27)
Exercise: Positivity of classical relative entropy
Prove that the classical relative entropy is non-negative. That is, prove (18.19), assuming ρ and σ are diagonal.
Exercise: Mutual information practice
Consider a 2-qubit system. First, calculate the mutual information of the two bits in
the classical mixed state
ρ = 12 (|00ih00| + |11ih 11|) .
(18.28)
This is a clearly a state with the maximal amount of classical correlation — if we
measure one bit, we know the value of the second bit.
Now, what is the maximal amount of mutual information for a quantum (pure or
173
mixed) state of 2 qubits? Write an example of a state with this maximal amount of
mutual information. (Quantum states with more mutual information than is possible
in any classical state are sometimes called supercorrelated.)
Exercise: Purification and the Triangle Inequality
Use strong subadditivity to prove the following identities for a tripartite system:
SA ≤ SAB + SBC
(18.29)
SA ≤ SAB + SAC
(18.30)
SAB ≥ |SA − SB |
(18.31)
Hint: Purify the tripartite system that appears in strong subadditivity by adding a
4th system, D, with ABCD in a pure state. This is always possible.
174
19
Entanglement Entropy in Quantum Field Theory
So far we have discussed entanglement in ordinary quantum mechanics, where the
Hilbert space of a finite region is finite dimensional. Now we will discuss geometric
entanglement entropy in quantum field theory. Space (not spacetime) is divided into
two regions, A and B, by a continuous curve:
(19.1)
This picture is at a fixed time. Region A is drawn as a circle, but for now it could be
any shape. (It could also be disjoint, but we will assume it is connected unless specified
otherwise.)
Quantum field theory is strictly speaking not bipartite,
HAB 6= HA ⊗ HB .
(19.2)
There are two things to worry about: first, in gauge theories, you cannot really localize
states. The gauge constraint is applied to the full system, so by looking at any subregion, you cannot decide whether it is a physical state obeying the constraint. This
issue (which also appears in ordinary quantum mechanical gauge systems) has been
addressed in some nice papers just in the last year or so, and we will ignore it entirely.
It turns out to not affect the discussion that follows very much.
The second issue is UV divergences. In a continuum QFT there are UV modes at
arbitrarily small scales across the dividing surface ∂A, and this makes it impossible to
actually split the full Hilbert space. To deal with this, we must impose a UV cutoff
175
by introducing the ‘lattice scale’ U V . With a finite cutoff, the Hilbert space of a
finite region is finite-dimensional, and most of the results of the previous section — in
particular, positivity of relative entropy and strong subadditivity — apply to QFT. In
the end we usually want to regulate the divergences somehow, but the leftover finite
pieces do not immediately obey the same properties, so we need to be careful about
tracking cutoff dependence throughout the problem.
19.1
Structure of the Entanglement Entropy
The divergent terms in SA come from UV physics. In the UV, any finite energy state
is the same as the vacuum state. Therefore to discuss the structure of the divergent
terms we can restrict to ρ = |0ih0|, the vacuum state of the full system.
UV divergences
The divergent terms depend on the theory and on the shape of region A. In a local
QFT, we expect the divergent piece to be a local integral over the entangling surface
∂A,
(div)
SA
√
dd−2 σ h F [Kab , hhab ] ,
Z
∼
(19.3)
∂A
where F is some (theory-dependent) functional of the extrinsic curvature and induced
metric on ∂A. This is for the same reason that when we do renormalization, we are
only allowed to add local counterterms to the Lagrangian; non-local terms come from
IR physics.
Let’s organize (19.3) as an expansion in powers of Kab . Since Kab ∼ 1/LA , this is an
expansion in powers of the size LA . What sort of terms can appear? In a pure state,
(div)
SA = SB , an in particular SA
(div)
= SB
. The extrinsic curvature is K ∼ ∇n with n
the unit normal; this flips sign if we consider region A vs its complement, region B.
(div)
Therefore SA
(div)
= SB
implies that only even powers of Kab are allowed:
(div)
SA
∼ a1 Ld−2
+ a2 Ld−4
+ ··· ,
A
A
where ai depend on the theory but not on LA .
176
(19.4)
The leading term in (19.4) is a UV divergence proportional to Area(A). This makes
sense: UV modes entangled across ∂A give a divergent contribution, and the number
of these modes is proportional to the area.
General structure and universal terms
Now let us further assume the theory is scale invariant. In the vacuum state of a
scale invariant theory, the only scales in the problem are U V and LA . Therefore,
4−d
by dimensional analysis, a1 ∼ 2−d
U V , a2 ∼ U V , etc. Thus, allowing also for a finite
contribution, we find the general behavior of the entanglement entropy in a CFT. In
odd dimensions d:
SACF T
∼ bd−2
LA
U V
d−2
+ bd−4
LA
U V
d−2
d−4
+ · · · + b1
d−1
LA
+ (−1) 2 S̃ + O(U V ) , (19.5)
U V
and in even dimensions:
SACF T
∼ bd−2
LA
U V
+(−1)
+ bd−4
d−2
2
S̃ log
LA
U V
d−4
+ · · · + b2
LA
U V
2
(19.6)
LA
+ const + O(U V ) ,
U V
The difference between even and odd comes from the fact that the “1/0U V ” term that
would appear in even dimensions actually turns into a log divergence (just as it does
in Feynman diagrams). The powers of (−1) are inserted by convention.
In the vacuum state, the bi and S̃ depend on the theory, but not on LA or U V .
In a non-scale-invariant QFT, or in an excited state of a CFT, there are other scales.∗
So in general, S̃ depends on the theory, the shape, and the state ρtotal . Furthermore, S̃ is
universal in the sense that it does not depend on ambiguities in the choice of regulator.
For this reason it is sometimes called the renormalized entanglement entropy.
∗
†
This is sometimes confusing in the CFT case but obviously true: even in a scale invariant theory,
an excited state with lumps of stuff 1 meter apart is different from an excited state with lumps of
stuff 2 meters apart!
†
The fact that it is independent of regulator is clear in the even dimensional case, since it is the
coefficient of a log. It is less clear for the constant term odd dimensions, since evidently shifting
U V → U V + a would change the finite term. In practice it seems to be well defined in CFT for
reasons I won’t get into here, but I’m not sure about the non-conformal case.
177
Area vs volume terms
The leading UV divergence is always proportional to Area(A), in any state. In the
vacuum we do not expect any extensive contribution to S̃, but in a random excited
state, we expect
S̃ ∼ Volume(A) .
(19.7)
This is for the same reason that we argued for volume scaling in a random state of a
lattice system. In a highly excited random state, the IR modes that contribute to S̃
should all be highly entangled with the outside, and the number of such modes scales
with volume.
Example: 2d CFT in vacuum
As a simple example, consider a 2d CFT in the vacuum state of the full system. Space
is a line, and region A is an interval of length LA . In this case the entanglement entropy
can be computed exactly (we will do this calculation later in the course) with the result
SA =
c
LA
log
.
3
U V
(19.8)
Here c is the central charge of the CFT (which, remember, roughly speaking counts the
degrees of freedom). This agrees with the general formula in even spacetime dimensions
(19.6), with S̃ = 3c .
If we instead consider a highly excited state, then we can’t do the calculation in general,
but in cases where it can be done the result in a typical state scales as S̃ ∼ cLA .
19.2
Lorentz invariance
In a Lorentz-invariant QFT, the density matrix of a spatial region A must contain all
of the same information as the density matrix of a spatial region A0 that shares the
178
same causal diamond. That is, for this setup:
(19.9)
we must have
SA = SA0 .
(19.10)
In words, this is because if we know everything about A, we can time-evolve to learn
everything about A0 . In formulas, it is because the reduced density matrices are related
by (perhaps very complicated and nonlocal!) unitary operation,
ρA0 = U † ρA U .
179
(19.11)
20
Entanglement Entropy and the Renormalization
Group
Entanglement entropy is very difficult to actually calculate in QFT. There are only a
few cases where it can be done. So what is it good for? One answer is the relation
to quantum gravity, which we’ll get to later. Another answer is that entanglement
entropy has led to deep insights into the structure of QFT. It is a tool that is almost
orthogonal to the usual tools of QFT, and can be used to prove general facts about
QFT that, so far, cannot be proved using any other method. The most important
example is on the irreversibility of the renormalization group in d = 3. We’ll now take
a brief detour to describe this result and the relevance of entanglement, as pioneered
by Casini and Huerta. We restrict to Lorentz-invariant QFTs.
20.1
The space of QFTs
The renormalization group connects conformal field theories:∗
(20.1)
Starting with CF T 1 in the UV, we deform by a relevant operator and flow down to
CF T 2 or, depending on the deformation, perhaps CF T 3 in the IR. These CFTs might
be free, or trivial, as in QCD, which is an RG flow between a free theory in the UV
and a gapped (empty) theory in the IR. The IR fixed points may also have relevant
perturbations, so we can continue the process and flow to new theories. Two natural
questions are:
∗
Strictly speaking, it connects scale-invariant theories. It is widely suspected that scale invariant
QFTs are necessarily conformal, but this is proven only in 2d and in 4d under certain assumptions.
180
1. Which CFTs can flow to which other CFTs? For example, can the green flow in
the figure exist, connecting CF T 3 to CF T 4 ? Or should it flow from CF T 4 to
CF T 3 instead?
2. Can there by closed cycles, connecting the IR back up to the UV like the red
dotted flow in the figure?
The RG involves integrating out degrees of freedom, so it would be very strange to find
closed cycles! We expect that each time we do into the IR, we reduced the number of
degrees of freedom. To make this intuition precise has been a longstanding problem in
quantum field theory.
20.2
How to measure degrees of freedom
To make this precise we need to define ‘number of degrees of freedom.’
Free energy is no good
One ‘obvious’ guess fails. Let’s try to measure degrees of freedom by computing the
thermodynamic free energy, log Z. This can be computed by the Euclidean path integral on Rd−1 × Sβ1 . At a fixed point, dimensional analysis fixes
F (β) = −ctherm Vd−1 T d
(20.2)
where ctherm is a dimensionless number that we might guess counts degrees of freedom.
However, ctherm does not necessarily decrease along RG flows. An example is the flow
from the interacting critical point of N bosons in d = 3, to the Goldstone phase with
N − 1 free bosons.
So we need a more sophisticated measure of degrees of freedom. The correct measure
depends on dimension, as do known results about the irreversibility of the RG.
d=2: c-theorem
This case is the easiest and has been understood since the 80s, when Zamolodchikov
181
proved that the correct quantity to consider is the central charge c. Zamolodchikov’s
c-theorem states
cU V ≥ cIR
(20.3)
in a unitary, Lorentz-invariant RG flow.
c plays many roles in a 2d CFT: It appears in the Virasoro algebra, in the trace
anomaly, in the stress-tensor correlation functions, in the Casimir energy on a circle,
in the thermodynamic free energy, and in the groundstate entanglement entropy. In
higher dimensions, these different quantities can have different constants associated to
them, so it is not obvious how to generalize (20.3) to higher dimensions. The picture
that has emerged in the last few years (conjectured in even dimensions long ago by
Cardy) is that the correct quantity to consider is the partition function on S d . Exactly
how this works depends on the dimension.
d=3: F -theorem
The correct measure of degrees of freedom in a 3d CFT is
F = − log |ZS 3 | .
(20.4)
It can be shown that this is equal to the finite term in the entanglement entropy of
a spherical region. That is, let A be a ball of radius LA . In the vacuum state the
quantity appearing in (19.5) obeys
S̃ = F .
(20.5)
FU V ≥ FIR .
(20.6)
This quantity obeys the ‘F -theorem’,
This was proved by Casini and Huerta using entanglement methods, described below.
d=4: a-theorem
In even dimensions, the partition function on S d has a log divergence due from the
182
conformal anomaly. The coefficient of this log divergence is called a:
log ZS 4 ∼ a log
R
U V
.
(20.7)
The same number appears in the entanglement entropy of a spherical region. In the
notation of (19.6),
S̃ ∝ a .
(20.8)
aU V ≥ aIR .
(20.9)
This obeys the ‘a-theorem’,
20.3
Entanglement proof of the c-theorem
Zamolodchikov derived the c-theorem in d = 2 using standard QFT methods, without
reference to entanglement entropy. Later, it was derived using entanglement entropy by
Casini and Huerta. Their derivation is very elegant, and exemplifies how entanglement
inequalities can be applied in QFT. Unlike Zamolodchikov’s proof, it also generalizes
to d = 3.
We consider any Lorentz-invariant QFT in 2d. Consider two boosted, overlapping
intervals A and B, arranged as follows:
(20.10)
We have also labeled the regions X, Y, Z. All of these are spacelike regions. Comparing
causal diamonds, Lorentz invariance, as discussed in section 19.2, implies
SA = SX∪Y ,
SB = SY ∪Z
(20.11)
and
SA∪B = SX∪Y ∪Z ,
183
SA∩B = SY .
(20.12)
Now, strong subadditivity implies
SA + SB ≥ SA∪B + SA∩B
(20.13)
SXY + SY Z ≥ SY + SXY Z .
(20.14)
i.e., (with ∪’s implied)
Parameterize the region lengths by r and R with
√
`(A) =
rR,
`(Y ) = R .
(20.15)
In the vacuum state, the entanglement entropy can depend only on the proper length
of the region. Thus SSA becomes
√
2S( rR) ≥ S(R) + S(r) .
(20.16)
Expanding with R = r + , this means
rS 00 (r) + S 0 (r) ≤ 0
(20.17)
or equivalently
C 0 (r) ≤ 0,
C(r) = rS 0 (r) .
(20.18)
(20.18) is the main technical result: the function C(r) is monotonic as a function of
interval size. Now for the interpretation. First, suppose our QFT is scale invariant. In
this case, from (19.8), the entanglement entropy is
Scf t (r) =
c
r
log
.
3
U V
(20.19)
Thus the Casini-Huerta C-function C(r) is proportional to the central charge at a
critical point,
Ccf t (r) ≡ rS 0 (r) =
c
.
3
(20.20)
Now, if the QFT is not scale invariant, then it describes an RG flow between some
UV CFT and some IR CFT. That is, the QFT at very short distances is equivalent
to CF TU V , and the QFT at very long distances is CF TIR . We are interpreting the
physical distance r as the RG scale. But we know that at the fixed points, C(r) is
184
given by the central charge,
C(r → 0) =
cU V
,
3
C(r → ∞) =
cIR
.
3
(20.21)
Integrating the equation C 0 (r) ≤ 0 from short to long distances,
Z
∞
drC 0 (r) ≤ 0 .
(20.22)
0
This proves the c-theorem,
cU V ≥ cIR .
(20.23)
Note that nowhere in this proof have we used the concept of a quantum field!!! We
used only locality, Lorentz invariant, quantum mechanics, and unitarity (in the guise
of the SSA inequality).
20.4
Entanglement proof of the F theorem
Casini and Huerta’s proof of the F theorem d = 3 is quite similar. In this case, there
is no other known way to prove that RG flows are irreversible – standard field theory
methods in even dimension rely on the conformal anomaly, which does not exist in odd
dimensions.
We will just briefly sketch the argument, since it is similar to d = 2. In a 3d CFT in
vacuum,
SACF T ∼
r
U V
− S̃
(20.24)
where S̃ is a constant, independent of r and U V . Therefore a natural guess for the
monotonic function is
F (r) = rS 0 (r) − S(r) ,
(20.25)
which agrees with S̃ at a critical point,
FCF T = S̃ .
(20.26)
To use SSA, we use a more clever version of the boosted-interval setup. Two boosted
185
balls, won’t work, because the union of the causal domain of two boosted balls is not
the causal domain of any ball. Instead we must arrange an infinite number of boosted
regions. Projected onto a single time slice, they look like this:∗
(20.27)
An argument similar to 2d implies that F 0 (r) ≤ 0, which establishes the F -theorem:
FU V ≥ FIR .
∗
Figure taken from Casini and Huerta 1202.5650.
186
(20.28)
21
Holographic Entanglement Entropy
21.1
The formula
We now turn to entanglement entropy in CFTs with a semiclassical holographic dual.
That is, we assume the CFT has a large number of degrees of freedom Ndof 1 (so
that `AdS `P lanck ) and a sparse low-lying spectrum (to suppress higher curvature
corrections, i.e., `AdS `string ). We also assume that the CFT is in a state ρ with
a geometric dual. This last assumption is needed since even in a holographic CFT,
not every state corresponds to a particular geometry (consider, for example, a linear
superposition of two black hole microstates with very different energies).
The entanglement entropy in this case is given by the holographic entanglement entropy
formula:
SA =
area(γA )
,
4GN
(21.1)
where γA is a codimension-2, spacelike extremal surface in the dual geometry, anchored
to the AdS boundary such that
∂γA = ∂A .
(21.2)
An extremal surface is a surface of extremal area. This looks roughly as follows, with
z the radial direction in AdS:
(21.3)
γA lives in a particular spacelike slice, so that is what is drawn here with the orthogonal
(time) direction suppressed.
Two additional comments: First, in (21.1), we are only allowed to include extremal
surfaces γA which are homologous (continuously deformable) to region A. Second, if
187
there are multiple extremal surfaces satisfying the homology condition, then the rule
is to apply (21.1) to the one which has minimal area.
History and Nomenclature
In the static case, this is generally referred to as the Ryu-Takayanagi formula, after
the authors who conjectured it in 2006. In time-dependent geometries it’s called the
HRT formula, after Hubeny, Rangamani, and Takayanagi. Important refinements, discussed below, were also made by Headrick, and many others. The static formula, and
the time-dependent formula in certain special states, were derived from the AdS/CFT
dictionary Zcf t (M ) = Zgrav (bdry = M ) by Lewkowycz and Maldacena in 2013. Because RTHHRTLM is a mouthful, I will refer to the general formula (21.1) as the HEE
(holographic entanglement entropy) formula.
Static case
In a static geometry there is a natural t coordinate, and symmetry implies that γA will
always lie within a fixed-t slice. An extremal surface in a fixed-t slice is the same as
a ‘minimal area surface’ inside this slice, so in this case the HEE formula reduces to
finding a minimal-area surface in a d − 1-dimensional space geometry.
Extremal surfaces are minimal-area with respect to deformations inside a fixed-t slice,
but maximal-area with respect to deformations in the t direction (since we can always
reduce the area of a surface by making it ‘more null’). The same is true of spacelike
geodesics, which extremize the length of a curve in spacetime, rather than minimizing
or maximizing it.
21.2
Example: Vacuum state in 1+1d CFT
Consider a 2d CFT in vacuum. Let region A by an interval of length LA ,
x ∈ [−
LA LA
,
].
2 2
188
(21.4)
The dual geometry is empty AdS3 , with metric
ds2 =
`2
(−dt2 + dx2 + dz 2 ) .
2
z
(21.5)
z is the radial direction, with the boundary at z = 0.
The state is static, so we can set t = 0. A codimension-2 extremal ‘surface’ in AdS3 is
one-dimensional, i.e., a geodesic. So the HEE formula instructs us to find a spacelike
geodesic, in the space geometry
ds2 =
`2
(dx2 + dz 2 ) ,
z2
(21.6)
connecting the points
P1 = (z1 , x1 ) = (0, −
LA
LA
) and P2 = (z2 , x2 ) = (0,
).
2
2
(21.7)
However, a geodesic that reaches the boundary like this will have infinite length, since
R dz
= ∞. This is the gravity dual of the statement that entanglement entropy in QFT
z
is UV divergent. To regulate the divergence, we follow the same procedure we used to
regulate the on-shell action, or holographic correlation functions: cut off the spacetime
at z = U V .
Thus we want to compute the length of this curve:
(21.8)
Parameterizing the curve (x(λ), z(λ)) by z, the regulated geodesic length is
Z
Length =
ds
Z
zmax
= 2LA
189
dz p 0 2
x (z) + 1
z
(21.9)
The factor of 2 is because we the geodesic goes out, and comes back, and we will only
integrate z ∈ [, zmax ] once. Treating (21.9) as a 1d “action”, it is easy to show that
the geodesic is a semicircle,
x=
LA
cos λ,
2
z=
LA
sin λ,
2
λ∈(
,π −
).
LA
LA
(21.10)
Plugging this back into (21.9) and doing the integral gives
Length = 2LA log
LA
U V
.
(21.11)
Therefore, applying the HEE formula (21.1),
LA
SA =
log
2GN
LA
U V
.
(21.12)
The map between gravity parameters and CFT parameters in AdS3 /CFT2 is
c=
3`
,
2GN
(21.13)
where c is the central charge, so
c
SA = log
3
LA
U V
.
(21.14)
This agrees perfectly with our general discussion of the structure of entanglement
entropy in QFT in even spacetime dimensions, (19.6), including the UV divergence.
The prefactor also agrees exactly with the known result in 2d CFT, (19.8).
Exercise: 2d HEE
Fill in all the missing steps — i.e., solve for the geodesic and do the length integral
— in the derivation of (21.14). Don’t forget to use conserved quantities to efficiently
solve the geodesic equation.
190
Exercise: Strips in d dimensions
Compute the holographic entanglement entropy of an infinite strip of width LA in a
d-dimensional CFT in the vacuum state. That is, with CFT coordinates (t, x, ~y ), region
A is the region x ∈ [−LA /2, LA /2], t = 0, ~y =anything.
21.3
Holographic proof of strong subadditivity
The proof of the strong subadditivity inequality in quantum mechanics is rather technical and tricky. The holographic proof, in static states, is easy! The statement of SSA
for a tripartite system is
SABC + SB ≤ SAB + SBC .
(21.15)
Let’s draw the various minimal-area surfaces:
(21.16)
We’ve picked a color scheme to reorganize the inequality a bit, so now it says
red + green ≤ blue + black .
(21.17)
The fact that the curves are minimal area immediately implies
red ≤ blue,
green ≤ black
(21.18)
so SSA follows.
This argument has also been extended to the time-dependent case. It is much trickier,
since the extremal surfaces need not all lie in the same spatial slice.
191
21.4
Some comments about HEE
HEE and Bekenstein-Hawking
The HEE formula is a generalization of the Bekenstein-Hawking area law for black
hole entropy. To see this, let’s apply the formula to a static black hole spacetime, and
choose region A to be all of space. In this case the boundary condition (21.2) on the
extremal surface is
∂γA = Ø .
(21.19)
It is tempting to say γ is the empty set, but this would not satisfy the homology
condition. A spatial slice of the black hole spacetime looks like
(21.20)
This is not simply connected, and the ‘empty set’ curve is not deformable to region A.
So, in fact, we must choose γA to be the horizon itself (which is extremal). Thus
SA =
area(horizon)
4GN
(21.21)
in agreement with Bekenstein-Hawking.
But why is this ‘entanglement entropy’ ? Actually, it might not be. More accurately, the
HEE formula computes the von Neumann entropy of the reduced density matrix, SA =
− tr ρA log ρA . This von Neumann entropy may or may not come from entanglement —
we can’t tell the difference without knowing the full system. In the black hole spacetime,
the ordinary thermal entropy is the von Neumann entropy of the thermal state ρ =
e−βH , so the HEE formula applied to the full space gives the thermal entropy. You can
also think of this as actual entanglement entropy coming from the entanglement of the
CFT with the thermal double.
192
Some words
Entanglement entropy is a measure of how quantum information is spatially organized
in a quantum state. In a general QFT, it is extremely complicated, and we do not
expect any tractable simple formula. The fact that it simplifies, and becomes geometric,
in holographic CFTs is a deep fact about strongly coupled systems. It means that the
organization of quantum information approaches a sort of simplified, universal limit at
strong coupling. How this happens and exactly how it is related to emergent geometry
is an unsolved, and presumably very important, problem in current research.
193
22
Holographic entanglement at finite temperature
In this section we will discuss the HEE computation in a black hole spacetime. For
explicitness, we will talk about the BTZ black hole in AdS3 , but all of the results hold
qualitatively in higher dimensions, too.
The BTZ metric is static, so we need only the fixed-time metric (` = G = 1)
ds2 =
dr2
+ r2 dφ2 .
2
r − 8M
(22.1)
This is dual to a finite-temperature state ρ = e−βH with temperature
T =
√
8M /2π .
(22.2)
Choose region A to be the boundary segment
A:
φ ∈ (0, R) .
(22.3)
So the figure is
(22.4)
There are many geodesics connecting the endpoints of region A. In fact there are an
infinite number, labeled by the integer number of times that the geodesic winds the
black hole. If R 2π, then there is one that is obviously the shortest, which does not
wind the black hole. This is the one drawn in the figure. The length of this geodesic
is infinite, but if we impose a cutoff at r = 1/U V , the resulting entropy is
(0)
(0)
SA
length(γA )
4GN
c
1
=
log
sinh(πRT ) ,
3
πT U V
=
194
(22.5)
with c = 3`/2GN .
For 0 < R 2π, this is the final answer. For R > 2π, the same formula (22.5)
computes the length of a geodesic that winds (possibly multiple times) around the
horizon. The winding geodesics do not satisfy the homology condition, i.e., they cannot
be continuously deformed to A. But we must also consider disjoint geodesics. For
example, the horizon itself is a geodesic H. This can be added to to the wrapped
geodesic γ (1) . The union
γ (1) ∪ H
(22.6)
(1)
is homologous to region A, if we view the orientation of H as opposite that of γA .
If region A is large, we must choose the dominant (minimal area) surface. We are
choosing between the red (wrapping) and blue (disjoint) surfaces in this figure:
(22.7)
The length of the red curve is what we computed above,
SAred
1
c
= log
sinh(πRT ) .
3
πT U V
(22.8)
The length of the blue curve gets a contribution from the short part, and a contribution
from the horizon:
SAblue
c
1
c
= log
sinh(π(2π − R)T ) + 2π 2 T
3
πT U V
3
(22.9)
The first term is the answer above, applied to AC . The second term is the thermal
entropy, which we know is area(horizon)/4.
The final answer is
SA (R) = min SAred ,
195
SAblue
(22.10)
The two contributions exchange dominance at some point R∗ > 2π. At this point
there is a sharp transition in the behavior of the entanglement entropy. The plot as a
function of system size is something like the solid line in this figure:
(22.11)
Pure vs mixed
Clearly with a black hole, SA 6= SAC due to the homology condition. In fact we found
(for R > R∗ )
SA = SAC + Sthermal .
(22.12)
Since Sthermal is the von Neumann entropy of the full space, this can be written
SA = SB + SAB
(22.13)
Thus the entanglement entropy of the black hole saturates the Araki-Lieb triangle
inequality,
SAB ≥ |SA − SB | .
This is a special feature of thermal states in holographic systems.
196
(22.14)
22.1
Planar limit
The infinite-volume limit of the CFT is a limit of the black hole where the horizon
becomes planar. We can take this limit for BTZ by assuming
T R 1,
R 2π .
(22.15)
In this limit, our answer (22.5) reduces to
c
SA ≈ πT R + subleading
3
(22.16)
This is equal to the thermal entropy density,
SA ≈ Sthermal (β) ×
R
.
2π
(22.17)
This makes sense: in the thermodynamic limit, the state is very mixed up, and the
subsystem itself just looks thermal at temperature β.
Geometrically, the reason behind (22.17) is that, in this limit, the extremal surface
‘hugs’ the black hole horizon for most of its length:
(22.18)
The contribution from the horizon is proportional to the horizon area, i.e., to the
thermal entropy.
We have discussed BTZ, but the same feature generalizes to planar horizons in higher
dimensions: for LA β, the extremal surface hugs the black hole horizon, giving a
volume-law contribution to the finite-temperature entanglement entropy.
197
23
The Stress Tensor in 2d CFT
In the last few lectures, we will go into more depth on the AdS3 /CFT2 correspondence.
First we need to cover some more ground in the basics of 2d CFT.
References: For 2d CFT I recommend: Chapter 2 of Polchinski’s text; Chapter 4
of Kiritsis’s text; the big book of Di Francesco et al; and the string theory lectures
notes by David Tong, available online. For the optimal introduction to the subject, I
recommend working through the chapter of Tong’s notes first, then working through
chapters 4-6 of Di Francesco et al.
23.1
Infinitessimal coordinate changes
Recall that in two dimensions, with complex coordinates in Euclidean R2 ,
ds2 = dzdz̄ ,
(23.1)
conformal transformations are holomorphic coordinate changes:
w = w(z),
w̄ = w̄(z̄) .
(23.2)
The coordinate change
z 0 = z + (z)
(23.3)
ζ µ ∂µ = −(z)∂z
(23.4)
φ0 (z 0 , z̄ 0 ) = φ(z 0 , z̄ 0 ) + ζ µ ∂µ φ
(23.5)
corresponds to the vector field
They act on fields as
The infinitessimal generators can be taken as
ζn = −z n+1 ∂z ,
ζ̄n = −z̄ n+1 ∂z̄ .
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(23.6)
The conformal generators make an algebra
[ζm , ζn ] = (m − n)ζm+n ,
(23.7)
and similarly for the barred generators. This is called the ‘Witt algebra’ (or centerless
Virasoro algebra).
The algebra is infinite-dimensional. There is one subalgebra, which consists of the 3
generators ζ0,1,−1 . These make the global subalgebra SL(2):
ζ−1 , ζ0 , ζ1
and ζ̄−1 , ζ̄0 , ζ̄1
⇒
SL(2, R) × SL(2, R) ∼ SO(2, 2) .
(23.8)
It is called the global subalgebra because these are the only ζn0 s that are non-singular
on the Riemann sphere. To see this, first look near z ∼ 0. Clearly the vector field
ζn = −z n+1 ∂z is regular only for n ≥ −1. Now, do the coordinate change w = 1/z,
and look near w ∼ 0. The vector field becomes
ζn = −z n+1 ∂z = −w−n−1 (−w2 )∂w
(23.9)
which is regular only for n ≤ 1. So the generators in (23.8) are the only ones regular
at both poles of the Riemann sphere.
23.2
The Stress Tensor
Translation invariance implies the action S is invariant under xµ → xµ +µ , for constant
µ . The classical stress tensor is defined by applying the Noether to this symmetry.
Promoting µ to an arbitrary function of xµ and varying the action must give something
proportional to ∂,
Z
δS = −2
√
d2 z gT µν ∂µ n .
(23.10)
This defines the stress tensor∗
4π δS
Tµν = − √
.
g δg µν
∗
(23.11)
Note the extra −2π compared to our when we computed gravitational stress tensors a while back.
This is just a convention but will be important to remember when we compare the two.
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It’s conserved
The Noether procedure guarantees that
∇µ T µν = 0 .
(23.12)
In a flat background (23.1), in complex coordinates the two components of this equation
are
¯ zz = 0 ,
∂T
∂Tz̄z̄ = 0 .
(23.13)
∂¯ = ∂z̄ .
(23.14)
where we have introduced the shorthand
∂ ≡ ∂z ,
From (23.13), Tzz is a holomorphic function of z, and Tz̄z̄ is anti-holomorphic. These
will be denoted
T (z) ≡ Tzz ,
T̄ (z̄) = Tz̄z̄ .
(23.15)
And traceless
So far we have used only translation invariance. In a theory that is classically scale
invariant, we can also conclude that the trace of the stress tensor vanishes. The symmetry in this case is the infinitessimal rescaling
xµ → xµ + λxµ .
(23.16)
∂µ ν = λgµν ,
(23.17)
Since under a rescaling
the variation of the action is
Z
δS ∝
√
d2 z gλTµµ .
(23.18)
Scale invariance implies that this integral vanishes; conformal invariance, or local scale
invariance, means we can make λ → λ(z, z̄) so Tµµ = 0. In complex coordinates,
Tzz̄ = 0 .
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(23.19)
This is the classical stress tensor. Even if it is traceless, the quantum stress tensor might
have a non-zero trace, for two different reasons. First, the UV regulator introduces a
scale, and may introduce a trace. In fact, in a renormalizable theory,
Tµµ (x) =
X
βgi Oi (x) ,
(23.20)
i
where Oi are the relevant operators of the theory, gi are the corresponding couplings,
and βg are their beta functions. So, for example, in massless QCD, although the
classical stress tensor is traceless, the quantum stress tensor has a contribution from
the non-zero QCD beta function. In a CFT, all the beta functions are exactly zero, so
the equation
Tµµ = 0
(23.21)
is true as an operator statement. The phrase “as an operator statement” means the
equation is true inside correlation functions, up to delta-functions where Tµµ hits other
operator insertions (we’ll see some of these delta functions below).
The second origin of a non-zero trace is a quantum anomaly. This happens even in
CFT, if we place the theory on a curved background. This is called the Weyl anomaly
and is important but we probably won’t have time to cover it.
Noether currents for conformal symmetries
T (z) is the Noether current for translations. That is, the current J µ with
J z̄ = T (z)
(23.22)
is the current associated to z → z+const. What are the Noether currents associated
to the general conformal transformation z → z + (z)? These are simply
J z̄ = (z)T (z) .
(23.23)
This is sort of obvious; to reproduce it from the Noether procedure, you can promote
(z) → (z, z̄) and apply the usual Noether procedure.
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As expected, the current is conserved,
¯ )=0.
∂µ J µ = ∂(T
23.3
(23.24)
Ward identities
Ward identities are the quantum version of the Noether procedure.
Suppose we have a general symmetry φ0 = φ + δφ. The fact that this is a symmetry
means the action and the path integral measure are invariant,
S[φ0 ] = S[φ],
Dφ0 = Dφ .
(23.25)
Thus, promoting → (xµ ),
Z
Dφe
−S[φ]
Z
=
0
Dφ0 e−S[φ ]
Z
(23.26)
R
µ
Dφe−S[φ]− J ∂µ Z
Z
=
Dφ(1 − J µ ∂µ )e−S[φ]
=
(23.27)
(23.28)
The first line is just renaming a dummy variable; the second line defines the current,
J µ , which may have contributions from both the classical action and the measure; and
the third line expands to linear order. It follows that
Z
h
J µ ∂µ i = 0
(23.29)
for all , and so
h∂µ J µ i = 0 .
(23.30)
This is the quantum version of the Noether procedure. The same exact steps, starting
R
instead with DφO1 (x1 ) · · · On (xn )e−S[φ] can be used to show that
h∂µ J µ (y)O1 (x1 ) · · · On (xn )i = 0 if xi 6= y .
(23.31)
The restriction to xi 6= y is necessary for the derivation to work. Just set the support of
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to a small circle around y that does not include any of the other operator insertions,
and repeat the steps above.
The equation (23.31) means
∂µ J µ = 0
(23.32)
This is an operator equation, ie it holds inside correlators, up to delta functions.
If there are insertions that collide with ∂µ J µ , we have to be more careful. Suppose x1 =
y. Then when we do the transformation inside the path integral, the transformation
also affects this operator insertion,
O1 → O1 + δO1 .
(23.33)
So now,
Z
−S[φ]
DφO1 (x1 ) · · · On (xn )e
Z
Z
Dφ(1 −
=
J µ ∂µ )(O1 + δO1 )O2 · · · On e−S[φ]
(23.34)
and the conservation law is modified to (restoring some dropped constants)
1
−
2π
Z
∂µ hJ µ (y)O1 (x1 ) · · · On (xn )i = hδO1 (x1 )O2 (x2 ) · · · On (xn )i,
(23.35)
D(y)
where D(y) is a disk enclosing y, where we’ve chosen to be non-zero. Allowing for
any of the operators to collide with the current, (23.35) becomes
∂µ hJ µ (y)O1 (x1 ) · · · On (xn )i =
X
δ(y − xi )hO1 · · · δOi · · · On i .
(23.36)
i
(23.36) is called the Ward identity.
As a residue
In two dimensions, using complex coordinates we can write (23.36) in a nice way. By
Stokes, we have
Z
µ
I
∂µ J ∼
D(y)
(Jz dz − Jz̄ dz̄)
y
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(23.37)
and
i
2π
I
dzJ(z)O(x) = −resz∼x J(z)O(x) .
(23.38)
x
Therefore, the Ward identity (23.36) can be written as the operator equation
δO(x) = −resz∼x [J(z)O(x)] .
(23.39)
Conformal Ward Identities
The Noether current for the conformal symmetry z → z + (z), from (23.23), is
J(z) = (z)T (z) .
(23.40)
Therefore the Ward identity for conformal transformations, allowing for both holomorphic and anti-holomorphic transformations, is
δ,¯ O(x) = −resz∼x [(z)T (z)O(x)] .
23.4
(23.41)
Operator product expansion
The product of two local operators can be Taylor-expanded as local operators at a
single point:
O1 (x)O2 (y) =
X
f12j (x − y)Oj (u) .
(23.42)
j
The sum is over all local operators in the theory. This is called the operator product
expansion (OPE). The OPE coefficients f12j (x − y) depend on the theory, but they
are restricted by conformal invariance, so that in fact the fijk ’s of primary operators
determine the fijk ’s of descendants as well.
The Ward identity, in the form (23.41), means that the singular terms in the O(x)T (y)
OPE are related to the conformal transformations of O.
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Primaries
Recall that primary operators transform as
0
0
0
O (z , z̄ ) =
dz 0
dz
−h dz̄ 0
dz̄
−h̄
O(z, z̄)
(23.43)
where (h, h̄) are called the conformal weights of O. The corresponding infinitessimal
transformation is
δ,¯ O(z, z̄) ≡ O0 (z, z̄) − O(z, z̄)
¯ + ¯∂O)
¯
= −(hO∂ + ∂O) − (h̄O∂¯
(23.44)
(23.45)
Comparing to (23.41), this means
T (z)O(w, w̄) ∼
hO(w, w̄) ∂O(w, w̄)
+
.
(z − w)2
z−w
(23.46)
The symbol ‘∼’ means that we only written the singular terms in the OPE; there is
also an infinite series of contributions regular at z = w. To check this gives the correct
residue, expand
(z)T (z)O(w) = T (z)(w)O(w) + (z − w)T (z)0 (w)O(w) + · · ·
(23.47)
and look at the simple pole in (23.46).
Upshot
The lesson is that singular terms in the T φ OPE contain exactly the same information
as the transformations of φ under conformal symmetry. The conformal algebra is one
and the same as the data in the singular part of the OPE.
23.5
The Central Charge
Now we will examine the transformation of the stress tensor under conformal symmetry, or equivalently, the T T OPE. Much of the discussion in this section is easiest to
understand first using an example like a free scalar field. This example is worked in
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every reference so I will not repeat here but encourage the reader to jump to the free
scalar section of Polchinski, Kiritsis, or Tong’s notes before continuing.
The stress tensor is not a primary, so we cannot plug O = T into (23.46). But, it does
behave similar to a primary under rescalings, with (h, h̄) = (2, 0):
T 0 (λz) = λ−2 T (z) .
(23.48)
This is basically dimensional analysis; the energy should have mass dimension 1, and
R
E ∼ T . Similarly, T (z̄) has scaling weights (h, h̄) = (0, 2). Requiring both sides of
the OPE to have the same scaling weight, it must have the form
T (z)T (w) =
c/2
X1 (w)
X2 (w)
X3 (w)
+
+
+
+ ··· ,
4
3
2
(z − w)
(z − w)
(z − w)
z−w
(23.49)
where c is a number (the 2 is inserted by convention), X1 is some field of dimension
(1, 0), X2 is dimension (2, 0), and X1 is dimension (3, 0). There are no terms more
singular than (z − w)4 , since fields must have positive scaling weight in a unitary
theory.∗ The exchange symmetry
T (z)T (w) = T (w)T (z)
(23.50)
set X1 = 0. The OPE is always invariant under permutations (in Euclidean signature).
One way to see this is that equal-time commutators in Lorentzian signature must
vanish by causality, and these equal-time commutators correspond to permutations of
the Euclidean correlator. This property is sometimes called locality or causality.
To fix the other two terms, we use the Ward identity for the scale symmetry (23.48).
The Noether current for scale symmetry is, from (23.23)
Jscale (z) = zT (z) .
(23.51)
Then using the Ward identity (23.41) gives our final answer for the singular terms in
the OPE:
T (z)T (w) ∼
c/2
2T (w)
∂T (w)
+
+
.
4
2
(z − w)
(z − w)
z−w
∗
(23.52)
I will not explain why, but this can be found in the references, in the section on the state-operator
correspondence and conformal reps. This also explains why c must be a number, not a field.
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The constant c is called the central charge. For a free boson, it turns out to be c = 1,
and for a free fermion c = 1/2. (see references).
Transformation law for the stress tensor
Knowing the T T OPE, we can use the Ward identity δT (w) = −resz∼w [(z)T (z)T (w)]
to find how T transforms under a conformal symmetry:
c 000
.
12
δT = −T 0 − 20 T −
(23.53)
The first two terms are the usual transformations of a primary. The last term is an
‘anomolous’ term coming from the central charge.
The finite transformation law, obtained by integrating (23.53) is
0
0
T (z ) =
where
dz 0
dz
−2 h
T (z) −
c 0 i
{z , z} ,
12
f 000 3 (f 00 )2
{f (z), z} ≡ 0 −
f
2 (f 0 )2
(23.54)
(23.55)
is called the Schwarzian derivative. The easiest way to derive (23.54) is to check that
it agrees infinitessimally with (23.53), and that it composes correctly under multiple
transformations.
Aside: Virasoro algebra
We won’t cover the operator formalism in this course, so I will not discuss the Virasoro
algebra in detail. Roughly, you can decompose the stress tensor into modes, Ln ∼
H
dzz n+1 T (z). Then the T T OPE (23.52), combined with the Ward identity, become
the Virasoro algebra
[Lm , Ln ] = (m − n)[Lm − Ln ] +
c
(m3 − m)δm,−n .
12
(23.56)
This is where the name ‘central charge’ comes from. It is another way of writing
(23.52): the T T OPE, the Schwarzian derivative, and the Virasoro algebra all contain
the same information.
207
23.6
Casimir Energy on the Circle
c is related to the Casimir energy of the theory on a circle. The mapping from the
plane to the cylinder of radius L is
z = e2πw/L .
(23.57)
The cylinder coordinate is identified w ∼ w + iL, since this takes us around a circle on
the z plane. Using the finite transformation law (23.54) gives
Tcyl (w) =
2π
L
2
z 2 Tplane (z) −
c ,
24z 2
(23.58)
where now we’re using ‘plane’ and ‘cyl’ to distinguish the stress tensor before and after
the transformation.
Let’s calculate the expectation value. On the plane, scale invariance sets all 1-point
functions to zero:
hTplane (z)i = 0 .
(23.59)
This is because the only scale-invariant function of z with weight 2 is 1/z 2 , but this
would not be translation invariant. Now using (23.58),
c
hTcyl (w)i = −
24
2π
L
2
.
(23.60)
This is a Casimir energy, in the usual sense: we started with a theory on a line (i.e.,
space is a line), then imposed periodic boundary conditions with period L, and found
energy ∼ 1/L. The size of the Casimir energy is fixed by the central charge. This is
our first indication that c might be a good way to the measure the degrees of freedom
of a CFT.
To compute the value of the energy explicitly, choose real coordinates w = τ + iφ,
where φ ∼ φ + L. The energy is defined in the usual way by integrating the stress
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tensor over a fixed-time slice:
1 L
∈ dxhTτcyl
τ (τ = 0, x)i
2π 0
Z L
1
cyl
=
dxhTww
+ Tw̄cyl
w̄ i
2π 0
Ecyl =
(23.61)
(23.62)
In the second line, we just did the change of coordinates w = τ + iφ, w̄ = τ − iφ, and
used tracelessness of the stress tensor, Tww̄ = 0. Evaluating this in vacuum gives the
Casimir energy
vac
Ecyl
=−
c 2π
.
12 L
(23.63)
In a general state, (23.61) gives
Ecyl = ∆ −
where
1
∆=h
2πi
c
12
I
I
dzzT (z) +
(23.64)
dz̄ z̄ T̄ (z̄) i .
(23.65)
Recall that J = zT (z) is the Noether current for scale transformations. Therefore, the
rhs is the expectation value of the scale operator. In an eigenstate, ∆ is the scaling
dimension of that state. So (23.64) says that the energy on the cylinder is the scaling
dimension on the plane, shifted by the central term. This makes sense, since time
translations on the cylinder correspond to scale transformations on the plane.
Exercise: free boson
The action of a free boson is S =
R
¯ Use the Noether procedure to find the
d2 z∂φ∂φ.
stress tensor. Then, compute the T T OPE and confirm that c = 1. (See Polchinski,
Kiritsis, Di Francesco, or Tong’s notes if you get stuck).
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24
The stress tensor in 3d gravity
Now we will compare the stress tensor of 3d gravity to our results of the previous
section. Consider the asymptotically-AdS3 metric
`2 2 r 2
ds = 2 dr + 2 dzdz̄ + hzz dz 2 + hz̄z̄ dz̄ 2 + 2hzz̄ dzdz̄ .
r
`
2
(24.1)
We have not written every possible term in the perturbation hµν , but it turns out that
other terms can be removed by a diff. Also, we will keep only the leading term in hµν
at large r, i.e., near the boundary, so we can assume that hµν is independent of r.
The Einstein equations imply that the perturbation is traceless and conserved:
hzz̄ = 0
(24.2)
¯ zz = 0 .
∂hz̄z̄ = ∂h
24.1
(24.3)
Brown-York tensor
The stress tensor of this geometry was computed in an exercise from one of the early
lectures. Let’s briefly review how this works. The Brown-York stress tensor is∗
on−shell
4π δSEinstein
Tij ≡ − √
g δg ij
1
1
= −
Kij − Kgij − gij .
4
`
(24.4)
(24.5)
The first two terms came from varying the Einstein action plus the Gibbons-Hawking
boundary term. The last term comes from the counterterm, with the coefficient set
in order to make the answer finite as r → ∞. Plugging the metric (24.1) into (24.5),
using (24.2) and doing a lot of work, eventually
Tzz = −
1
hzz ,
4`
Tz̄z̄ = −
∗
1
hz̄z̄ .
4`
(24.6)
We’ve changed conventions by a factor of 2π compared to some earlier lectures. This is just a
choice, made to agree with our convention for the CFT stress tensor.
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Thus the Brown-York metric is traceless, conserved, and therefore holomorphic/antiholomorphic just like in CFT.
24.2
Conformal transformations and the Brown-Henneaux central charge
Under diffeos, the metric (24.1) transforms as
δgµν = Lζ gµν .
(24.7)
What vector fields ζ preserve the form of the metric (24.1)? The answer is
`4 00
¯ (z̄)
2r2
`4
z̄ → z̄ + ¯(z̄) − 2 00 (z)
2r
r
r 0
r → r − (z) − ¯0 (z̄)
2
2
z → z + (z) −
(24.8)
for arbitrary functions (z) and ¯(z̄). Near the boundary, these act on z, z̄ just like
conformal transformations. The extra ∂r piece acts as a rescaling.
Thus transformations of AdS3 that preserve the asymptotics of the metric coincide
with 2d conformal transformations!
Let’s set ¯ = 0 and focus on the holomorphic conformal transformations. Under (24.8),
the metric transforms as
`2 000
0
0
ds → ds + −2hzz − hzz + dz 2
2
2
2
(24.9)
Thus the dz 2 piece of the metric, which we interpreted as the gravitational stress tensor
up to a factor of −1/4`, transforms as
`
δ T = −∂T − 2T ∂ − 000 .
8
(24.10)
This is exactly the transformation law in 2d CFT derived in (23.53). Comparing the
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coefficient of the anomalous term, we see
c=
3`
2GN
(24.11)
where we’ve reinserted a factor of GN (previously set to 1) by dimensional analysis.
This is called the Brown-Henneaux central charge, after Brown and Henneaux who computed it way back in 1987 – long before AdS/CFT, and even well before the discovery
of the BTZ black hole or the Brown-York stress tensor. They used a different method,
based on conserved charges, which directly produces the Virasoro algebra (23.56) as
the asymptotic symmetry group. As far as I know, they did not recognize the relation
to the 2d conformal group.
24.3
Casimir energy on the circle
Recall the metric of the 3d black hole (BTZ):
ds2 = −(
r2
dr2
2
−
8M
)dt
+
+ r2 dφ2
2
2
2
`
r /` − 8M
(24.12)
It is up to us whether to identity φ ∼ φ + 2π (since there is no conical defect even if
we leave φ ∈ [−∞, ∞]). The BTZ black hole is the solution with φ ∼ φ + 2π. The
boundary of this spacetime is the Lorentzian (t, φ) cylinder, so this is dual to the CFT
on a cylinder.
In an exercise in a previous lecture you computed the energy of this solution, and found
E = M.
Global AdS can also be written in the form (24.12), by choosing M = − 8` . Therefore
the gravitational energy of the groundstate on the cylinder is
`
c
Evac = − = −
,
8
12
(24.13)
where c take the Brown-Henneaux value (24.11).
This is equal to the Casimir energy of a 2d CFT (23.63). It was actually guaranteed
to agree once we found the transformation law (24.10) agrees with CFT, because the
212
finite version of this infinitessimal transformation must be the Schwarzian derivative,
on the gravity side just as it was in CFT.
213
25
Thermodynamics of 2d CFT
In this lecture we will discuss the thermodynamics of 2d CFT, including the torus
partition function and the famous Cardy formula for CFT entropy. Later we’ll come
back to holography, but this note is entirely in CFT.
The partition function of a theory at finite temperature is
Z = Tr e−βH =
X
e−βE .
(25.1)
states
This is a sum over states in some Hilbert space; that Hilbert space depends on our
choice of the space in which the theory lives, i.e., our choice of boundary conditions
on the fields. If we choose space to be a circle, then
Z=
X
e−βEcyl =
states
X
c
e−β(∆− 12 ) ,
(25.2)
states
where ∆ is the scaling dimension of the state. The second equality comes from (23.64),
and assumes Lcyl = 2π.
As usual, the trace (25.1) is equal to a path integral in periodic imaginary time, with
period β. Since space is also periodic, this is a path integral on a torus, with a ‘space’
identification and a ‘thermal’ identification:
w ∼ w + L ∼ w + iβ .
(25.3)
Z(β) is equal to the Euclidean path integral on the torus (25.3).
Our aim is to compute this path integral at high temperature. First we discuss general
properties of CFT on a torus in the next couple subsections.
25.1
A first look at the S transformation
Consider a 2d QFT (not necessarily conformal yet) on a Euclidean torus. The most
general torus is specified by two lattice vectors ~v1 , ~v2 on the (tE , φ) plane, meaning that
214
we identify all points related by
(tE , φ) ∼ (tE , φ) + m~v1 + n~v2 ,
m, n ∈ Z .
(25.4)
If the theory is rotationally invariant (i.e., if its Lorentzian counterpart is Lorentz
invariant), then we may w.l.o.g. rotate ~v1 to lie on the tE axis. If the theory is scale
invariant, then we can also w.l.o.g. set its length to ~v2 = (0, 2π). Thus in a conformal
field theory, we are led to consider the theory on the torus
(tE , φ) ∼ (tE + β, φ + θ) ∼ (tE , φ + 2π) ,
(25.5)
where β and θ are arbitrary real numbers.
In the complex coordinate z =
1
(φ + itE ),
2π
τ=
the torus is specified by a complex number
1
(β + iθ) .
2π
(25.6)
The identifications are
z ∼z+1∼z+τ .
(25.7)
The path integral on this torus will be denoted Z(τ, τ̄ ). τ is called the modulus, or
complex structure modulus of the torus.
Converting to trace
The path integral on this torus can be converted to operator language by declaring
that φ is ‘space’ and tE is ‘time.’ Then we take states on a constant-tE surface and
evolve them along the vector β∂tE + iθ∂φ . Thus the path integral can be rewritten as
Z(τ, τ̄ ) = Tr
−βH+iθJ
H(0,2π) e
(25.8)
where J is the angular momentum (which, by convention, generates motion along −∂φ ,
hence the sign flip). In the subscript, we have noted explicitly what Hilbert space
to trace over: it is the Hilbert space of states defined on the spatial slice where φ ∈
[0, 2π] with tE held fixed. That is, the fields obey the boundary condition X(tE , φ) =
X(tE , φ + 2π). This is not standard notation; usually people just write ‘ Tr ’, with
H(0, 2π) implicit.
215
Large conformal transformations
We already know how conformal transformations act on the plane, as holomorphic/antiholomorphic coordinate changes. Now we want to examine conformal symmetry on the
torus. All of the usual conformal transformations z → z +(z) are still symmetries, but
there are also new, ‘large’ conformal transformations, which cannot be continuously
connected to the identity.
The S transformation
One way to see this is to re-slice the torus path integral in a different way. We’ll start
with an intuitive explanation in the simplest case θ = 0, then come back to the general
story below. The torus is Euclidean, so we are free to switch the roles of tE and φ
when we construct the trace by declaring tE is ‘space’ and φ is ‘time’. Then following
the usual logic, we find
Z(τ, τ̄ ) = Tr
H(β,0) e
−2πJ
.
(25.9)
Therefore we have written the same path integral in two different ways (25.8) and
(25.9),
Z(τ, τ̄ ) = Tr
H(0,2π) e
−βH
= Tr
H(β,0) e
−2πJ
.
(25.10)
This is true in any respectable quantum field theory. In a general (non-conformal)
QFT, the Hilbert spaces in these two expressions are not the same. The first is the
Hilbert space on the circle φ ∼ φ + 2π, and the second is the Hilbert space on the circle
tE ∼ tE + β. Now, in a scale-invariant theory, these are related by a rotation by 90◦
followed by a rescaling by 2π/β. Thus, in a CFT,
Tr
−2πJ
H(β,0) e
2
= Tr
− 4π
H
β
H(0,2π) e
.
Here the rotation by 90◦ sends J → H, and the rescaling inserts the factor of
It follows that
Z(β) = Tr e
−βH
=Z
4π 2
β
(25.11)
2π
.
β
.
(25.12)
This is called the S transformation. It came from a large conformal transformation
that swapped space and Euclidean time, and it relates the thermodynamics at high
temperatures to the thermodynamics at low temperatures.
216
25.2
SL(2, Z) transformations
The S transformation is one of an infinite group of large conformal transformations on
the torus. These come from all the different ways of slicing the torus. We can think of
these as ways of choosing the fundamental domain for the torus on the z-plane. The
usual fundamental domain is the tilted rectangle with
z ∼z+1∼z+τ .
(25.13)
But we can instead choose the even-more-tilted rectangle
w ∼w+1∼w+τ +1 .
(25.14)
This is precisely the same torus, since the lattice m + nτ with m, n ∈ Z is unchanged.
From the point of view of the z coordinate, the w coordinate system ‘winds’ around
the torus:
F IGU RE : T wotori, windingcoords
(25.15)
This coordinate transformation is called ‘T ‘. It acts on the modulus as
T :
τ →τ +1 .
(25.16)
The winding means that this coordinate transformation cannot be continuously deformed to the identity, i.e., there is no infinitessimal version.
Every theory, conformal or not, is invariant under T :
Z(τ + 1, τ̄ + 1) = Z(τ, τ̄ ) .
(25.17)
This is because we haven’t actually done anything, we’ve just rewritten the same torus
path integral in a different coordinate system. We can also check explicitly from the
trace formula:
Z(τ + 1, τ̄ + 1) = Tr e−βH+i(θ+2π)J .
(25.18)
For a theory with only bosons, this is equal to Z(τ, τ̄ ) since angular momentum is
integer quantized. In a theory with fermions, we have to be more careful about imposing
boundary conditions on the fermions, but in the end it still works.
217
There are in fact an infinite number of ways to slice the torus. The general choice of
lattice vectors v10 , v20 that generate the same lattice is

v10

v20


a b
=
c d


v1
v2

 ,
(25.19)
where
a, b, c, d ∈ Z

The matrix 
a b
and
ad − bc = 1 .
(25.20)

 is therefore an element of the group SL(2, Z)— i.e., , 2x2
c d
matrices with integer elements and unit determinant. This re-slicing maps
τ→
aτ + b
.
cτ + d
(25.21)
SL(2, Z)
 is generated
 by the S and T transformations. The S transformation is the
0 1
, which acts as
matrix 
−1 0
S:
τ → −1/τ .
(25.22)
Above, we discussed the S transformation with zero angular potential, τ = iβ/(2π).
In this case S : β →
4π 2
β
as claimed above.
Unlike T , the other SL(2, Z) transformations are not symmetries of a general QFT.
This is because the new space circle, which defines the Hilbert space, is not the same
as the old space circle. It is only in a scale-invariant theory that we can rescale these
two circles and see that they have the same Hilbert space. Thus in conformal field
theory, SL(2, Z) is a symmetry:
Z(τ, τ̄ ) = Z
aτ + b aτ̄ + b
,
cτ + d cτ̄ + d
218
.
(25.23)
25.3
Thermodynamics at high temperature
(We will restrict to the case θ = 0, but angular potential can be included at the expense
of slightly more complicated formulas.) First let’s calculate the partition function at
very low temperature:
Z(β) =
X
e−βE ≈ e−βEvac
(β → ∞) .
(25.24)
states
This is simply the statement that at very low temperature, the vacuum state dominates
c
the canonical ensemble. Above we found the Casimir energy Evac = − 12
, so
βc
Z(β) ≈ e 12
(β → ∞) .
(25.25)
Now, what about high temperatures? Modular invariance requires
Z(β) = Z(4π 2 /β) ,
(25.26)
so we can repeat the derivation at low temperature using the S-transformed partition
function. Replacing β → 4π 2 /β in (25.25) gives
π2 c
Z(β) ≈ e 3β
(β → 0) .
(25.27)
The corresponding free energy, defined by Z = e−βF , is
π2
F = − cT 2
3
V
2π
,
(25.28)
where we’ve written the formula for a general ‘volume’, which we’ve been assuming
is 2π. This is a remarkable formula. The scaling with the temperature in the free
energy is fixed by dimensional analysis in a CFT, but the coefficient is not. The hightemperature free energy dominated by very heavy states, and is usually impossible to
calculate in a strongly interacting QFT. But in this case, modular invariance relates it
to the Casimir energy of the vacuum state.
There are two immediate consequences. First, this confirms once again that c should
be interpreted as a measure of degrees of freedom. Second, it fixes the asymptotic
219
density of states. To see this, write the result as
X
−βE
e
Z
≈
dEρ(E)e−βE ≈ eπ
2 c/(3β)
(β → 0) .
(25.29)
states
The first sum is over all states of the theory; the integral is over energies, with ρ(E)
the density of states.
The rhs is very singular as β → 0. Each individual term on the lhs is regular at β = 0,
so the singularity can only come from the infinite sum. The strength of the divergence
must be related to the growth of ρ(E) as E → ∞.
To make this more quantitative, we can use standard thermodynamic formulae. The
thermodynamic entropy and energy are
2π 2 c
S = (1 − β∂β ) log Z =
3β
2
π c
E(β) = −∂β log Z = 2 .
3β
(25.30)
(25.31)
Putting these together, we have
r
S(E) = 2π
c
E.
3
(25.32)
As usual in stat mech, this formula should give the density of states via
ρ(E) = eS(E) .
(25.33)
As a check, let’s plug this into the partition function and see that it reproduces the
expected singularity as β → 0:
Z
Z(β) ≈
dEρ(E)e−βE
(25.34)
dE exp (S(E) − βE)
(25.35)
Z
≈
≈ exp (S(E∗ ) − βE∗ )
220
(25.36)
where E∗ is the saddlepoint value, defined by
S 0 (E∗ ) − β = 0 .
(25.37)
This saddlepoint is just the usual thermodynamic value of the energy. Using (25.32),
finding the saddlepoint, and plugging in, we find
Z(β) ≈ exp
π2c
3β
(25.38)
as claimed. (This is not a surprise; the whole point of thermodynamics and Legendre
transforms is to solve this saddlepoint equation for you.)
The entropy formula (25.32) is often called the Cardy formula. It applies to any CFT
as E → ∞.
Exercise: Cardy formula with angular potential
Derive the asymptotic density of states ρ(E, J) by applying modular invariance to the
partition function including angular potential, Z(τ, τ̄ ) = Tr e−βH+iθJ .
26
26.1
Black hole microstate counting
From the Cardy formula
The BTZ metric,
2
ds = −
r2
− 8M
`2
dr2
dt + 2 2
+ r2 dφ2
r /` − 8M
2
221
(26.1)
with φ ∼ φ + 2π, has energy E = M and entropy
area
4
1
2πρ+
=
4
π√
=
8M `
2 r
c
= 2π
E.
3
S =
(26.2)
In the last line we used the Brown-Henneaux central charge, c = 3`/2 (with GN = 1).
This exactly matches the Cardy formula, (25.32).∗ This match was found by Strominger in 1997. What is important about this result is that the Cardy formula counts
microstates in statistical mechanics. They are microstates in the dual CFT, but by
holographic duality, they must be microstates of quantum gravity in AdS3 as well.
We’ve just counted them without actually enumerating them, but a great deal of
progress has been made enumerating them as well.
26.2
Strominger-Vafa
Historically, the first black hole microstate counting was a string theory calculation by
Strominger and Vafa. It gave the same answer as (26.2). This calculation was important because it laid to rest any final doubts about whether black hole thermodynamics
is really thermodynamics (i.e., coming from stat mech) or just a mysterious analogy.
In quantum gravity, black hole entropy counts microstates:
SBH (E) ≈ log ρQG (E) ,
(26.3)
where ρQG is the density of states in quantum gravity. This is an extraordinary window
from low-energy physics into the theory of quantum gravity well above the Planck scale.
∗
Actually the Cardy formula was for E → ∞, whereas the black hole formula is for E > 0. It is
possible to get the match at all energies but requires further input from string theory, or some further
assumptions about the spectrum of the CFT.
222